I found more twin prime solutions. For $n = 396$, $d = 2$ I found four; the smallest is $x =$
$${\small
434219772837481616940726338933362452273916097301492635627291443512051997247961750150908315825206335025681182252954355463934702107964728269885096262688644034853639395077615105799339919571601786192665103151170947581386505769710635828116973131647706180301379657813220432413064536727826883282252811784121256116486385454859849292857777667719016881251956101283374338871503089431823276421261037909974359}$$
For $n = 420, d= 2$ I found $x =$
$${\small797632045320122442922746848370123218495083238021539262463793196558953236606673684986633691999805212011226776035396567202878987657452660659814874176096052099397205337415190293836597252999499063444222715671072742933640380698030812583452694556061309433747142801949662038045758937332990610602377203706320829077950947557183280793831442171456664369310820656043310649088552521802135329739094023210425721483954126295715977820771}$$
I wrote a Java program to find modular square roots using known factorizations of $10^n+1$ at https://stdkmd.net/nrr/repunit/10001.htm
The $n = 396$ solution shown may not be the second smallest twin prime solution because I did not analyse $10^n+1$ with unknown factorization (eg: $323, 392$).
Note that all these solutions involve only probable primes.
UPDATE:
I analyzed all $n \le$ 500 for which complete factorizations are known,
using Shank's $(ln\ 10^n)^2$ estimate for prime gaps: $2 \le d < 5.3n^2$. The following table shows the number of solutions of $n$ for each $d$.
\begin{array}{r|l|r} d & n & total \\
\hline 2 & 36 \ 396(4) \ 420 \ 468 & 7
\\ 4 & 2 \ 70 \ 150 \ 154 \ 210(27) \ 270(8) \ 298 \ 306(2) \ 330(216) \ 350(4) \ 390(10) \ 450(519) & 791
\\ 12 & 370(3) & 3
\\ 14 & 336(2) \ 396(3) & 5
\\ 54 & 72 & 1
\\ 76 & 150(2) \ 210(16) \ 238 \ 270(4) \ 306(2) \ 330(209) \ 350(3) \ 390(15) \ 450(464) & 716
\\ 82 & 288 \ 336 \ 396(5) & 7
\\ 136 & 370(2) & 2
\\ 478 & 336(2) \ 420(2) & 4
\\ 1364 & 350 \ 370 & 2
\\ \hline
\end{array}
I analyzed many of the remaining known factorizations ($10^{2021}+1$ is not one of them!). I found another twin prime solution: $n=1008$ with $d=2$, $x=$
$${\small415119285335713138107859159895921470120127901771012286705147863784377859054383205862088683121228843823644091024472578295715781891723489781341595609399956375971523325157129204031439607287157565203433091945798569106366453441681028958853009984899996962841286597261664058243897757317082387302022827796188228872367615940280481644558354868664735437056336615963301573812328050683875074515986286867031968871583403569829949341102058719072290611202615565955752049854880027669507042067569064851453754109081689314779674580036777493429088593987592215948631397812098103849006767378312184473327978598863061300578414744304808368322583101196999821431867760847600988156486022394149378510713398269801057016361947970128875578649362095899180615435692074486807012156095811778896443290518016088027948808773134635025220295845062105375107159001836188571569967805721691214024809680987944490924904786048603374165125520526174101682333287056313013054613391604106858759925794745594124906616469238525225716143856670923606283395248479131507}$$
I found three large solutions: $n=1530$ with $d=4$, $n=2442$ with $d=4$, and $n=3011$ (odd!) with $d=20690$ and $x=$
$${\small88690623974342445320706035164907895149666724533149745483024674929494064077161748069463728371848588950263928708885107056062087458873420997129551966175554795722180833514256206712208756791349423181129064370039115228768196330069003307522127855902035611295636718786724552834173077606816947565351851981240290592919330234435002605544818898704383050138705647019342509295291603847356454590383853903649034831518476190799174284971745361604565835545543751726529610775367997499873686078754015523170389723175091842178101899712480880871125974133817901518538674999703972616649632379652346395327832021456141800194701647932112291489233775928062470099635623529564608650310241193184759187003057364275380404785676729319256559060638811849744657325135099995176231775532397706618363314457193916318557002601459452270771049765689157482464270159966495073998419806308753511281083712198273207669118838966447628803262451478908503097089308456131000369654881518026667709557445311154935973975062432608109650859738547758359753161811236281693386411516599337580682444733887155249147703101490725039867545594773022646082904694935305608529996539856673800894879204151359288642251946537962882362542598385355877258435885255720463281832318996714749127620565928295871807381765069313332957959393036520560765777218866425812066806566968512664298253194097198110254106398916193178093323285510247757090049514423671929016532305341758972853095343072371415631421656596311059421635680366649419960217875499119738356663626923434263846557987005717753122760707664468827714474707911315796646846223047293968656604803860055531395007086544292734839551265327070525560544860595807571049301946018895129458512514361858289027536454436450967551887360811792658089837039529387070983777359590036217218745184175803793616945894809150630864349323615934382263207217535053701872322473625965756309360763380729005686068625698261693375747392352007831638590079855262236534947866527970435414431816193335704235728803592301557775355412534721854441649050543489915852763945114633640120859359425753981309635846954784400203166716706835000886558080563709914367339817642986577630512066137497993335131126095180539966000395250657557890723647715298981683035616063901855227374869260175148423588397623462842825972640026055333967889877654729507728851982076103225242572504527429707508826564462178655965426323365276645348503046435607191738892135058428362640796305615358118855629551341911741579489817482134866791604658301235483655504562219338888352571792484327753544399478907161167075036271824602275963945345494267916881834140919492205225308097367770069805455016950009471893642408322180438063500438940054744323617310086174876710854406021502917708656972234075774384689936433268632860789883740306973585819078135116827900294591791739792869006503475384136963506082899865375050762137583035828837915859970271078112397338135202345323724701287354928723862525723289692127491314630211880612187498071770083275673021172070970359804795314571223002210253991420896839561827296295193582310969508838235157649439819871296875917}$$
UPDATE (May 28, 2021)
I attempted to find solutions bigger than $n = 3011$, but found none!
I rewrote my program in C using GMP and ran it on AWS with 96 cores.
The following table summarises my results for all known complete factorizations
of $10^n+1$ for $n > 3011$.
For each $n$ the search was up to $d$ for the given ratio $\frac{d}{n}$.
Note that a ratio of $\mathbf{2.3}$ corresponds to the expected number of
primes in the range $[10^n,10^n+2.3n] = 1$,
a ratio of $\mathbf{96}$ corresponds to the current record prime gap
(of $8350$ for an $87$ digit prime number),
and a ratio of $\mathbf{6.9}$ corresponds to the biggest solution I found: $n=3011, d=20690$.
\begin{array}{r|l} n & \frac{d}{n} \\
\hline <22292 & 100
\\ 22292 & 21
\\ 22303 & 10
\\ 23592 & 53
\\ 23734 & 3
\\ 26014 & 10
\\ 32962 & 10
\\ 46957 & 3
\\ 47248 & 12
\\ 64439 & 1.0
\\ 80363 & 1.0
\\ 95594 & 1.0
\\ 103624 & 1.0
\\ 132586 & 1.0
\\ ^*180178 & 0.10
\\ ^*268207 & 0.10
\\ ^*1600787 & -
\\ \hline
\end{array}
$^*$For the last three values of $n$ the largest prime factor is a probable prime. I did not analyse the last value, because I estimate that a single calculation of $x^y$ mod $z$ with 1600000 digit numbers would take about 70 hours.
UPDATE (June 11, 2021)
For the largest known solution $(n=3011, d=20690)$ the primes $x$ and $x+d$ have been certified using Primo and recorded in factordb.com - search for primes with 3011 digits.
UPDATE (July 8, 2021)
I searched for all twin prime solutions $(d = 2)$ for $ n = 1008$ and found a total of 18. The largest, a new record, is $x=$
$${\small959366284693377033651389735182738458526276121193598553484439760604241337237451497452789960716688792639495153272015332792541955520436759663588757045933885829578173457125721695213851669376206161760331060025909588497751253712340327354707310139498467167235392925523839181171301134866240381669769835527505988247393867108561444781188402425283990574068760196925230932242847785476986116528280435669208099164158860477631087793039498552963323814612240444137080246220725308557204315700968442132019195862098047534857069372757341953852762143251682584867890954165891418336661720575672057626771116234845905889545915860780107445581724924411155250266226552503036267220337257336647317889646751823745301888424349108532006368071312231688002693274043002649314803790153000719816292711634980344850814644852432359001479449220906110949158858923721913836167918792291736846984222096022979367999907004169558381802601034072450079249418137015546418369167885051996435963305528417728081298205640144099322592183826280769897515421235519976901}$$
No more were found for $1008 < n \le 268207$.