We might consider a slightly more efficient method to identify a fundamental unit for the quadratic case.
Let us pick the integers in $\mathbb Q[\sqrt{7}]$. We know that any unit will solve one of the equations
$x^2+ax\pm1=0,$
and since $-1$ is nonquadratc $\bmod 7$ we can safely limit our trials to the $+$ sign in the constant term. By brute force or using continued fractions as a guide, we find $a=\pm16$ is the absolutely smallest candidate for which the roots are in the required field. Thereby we identify any $\pm8\pm3\sqrt7$ (independent choices of the signs) as a unit in this integer domain. That this is the fundamental unit in connected, in the quadratic case, with the coefficient $a$ being absolutely smallest among all nonzero possibilities giving roots in the required domain. Powers of this fundamental unit and their negatives correspond to absolutely larger $a=\pm254,\pm2048,...$.
The proposal is that a unit in the cubic domain will satisfy an equation of the form
$x^3+ax^2+bx+1=0$
where the odd degree allows us to specify the $+$ sign in the constant term; roots with the $-$ sign there are just negatives of those using the $+$ sign. We expect that raising any unit to a power will tend to increase the magnitudes of $a$ and $b$. On this argument a unit that minimizes $|a|+|b|$ is expected to be fundamental.
We try this with the integers of $\mathbb Q[\sqrt[3]{7}]$. One such unit is $-2+\sqrt[3]{7}$ (there is no $\sqrt[3]{49}$ term in this case) giving the polynomial equation
$x^3+6x^2+12x+1=0, |a|+|b|=18.$
Squaring this gives roots of
$x^3-12x^2+132x-1=0, |a|+|b|=144>18,$
and higher powers blow up the "norm" even further as with the quadratic case.
What if we try out roots of $x^3+ax^2+bx+1$ where $|a|+|b|\le18$? We have $685$ candidates:
$x^3-18x^2+0x+1=0\to\text{ casus irred.}$
$x^3-17x^2-1x+1=0\to\text{ casus irred.}$
$x^3-17x^2+0x+1=0\to\text{ casus irred.}$
(Casus irreducibilis equations will not have roots in a real integer field and are not further analyzed.)
...
$x^3-6x^2+9x+1=0\to x=2-\phi^{2/3}-\phi^{-2/3} (\phi=(1+\sqrt5)/2)$
...
$x^3-6x^2+12x+1=0\to x=2-\sqrt[3]{9}$
...
and it turns out that the only cases having irrational roots in the required domain are $x^3+6x^2+12x+1=0$ $x^3+12x^2+6x+1=0$, the latter derived from the former through multiplicative inversion.
If this reasoning is correct, then the fundamental unit in the integers of $\mathbb Q[\sqrt[3]{7}]$ is $\sqrt[3]7-2$ or any number derived from it by some combination of additive and multiplicative inversions.
The polynomial equation approach can be used systematically to search for units and identify units that would he fundamental accordingvyo the $|a|+|b|$ criterion above.
Start with the equation
$x^3+ax^2+bx+1=0$
where $a$and $b$ are integers. Without loss of generality we may take $a$ positive, $b<a$, and for roots in a real cube-root field the discriminant criterion for only one real root holds:
$-D=4(a^3+b^3)-a^2b^2-18ab+27>0.$
Thus for any choice of $a$ only a finite range of $b$ values are allowed, and we try out each of these to find cases where $(-D)/3$ is a perfect square. In these cases the root of the cubic will be a unit less than $-1$ in $\mathbb Q[\sqrt[3]{m}]$. The unit with minimal $a$ will then be fundamental, and its negative should correspond to the entry in the Project Euclid table given here.
For example, we try $a=12$. Then the discriminant criterion becomes
$-D=4b^3-144b^2-216b+6939>0,$
and the only values of $b$ less than $12$ meeting this condition are $-7,-6,-5,...+6$.
We render $-D$ for eachbof these trials and find two where the result is three times a perfect square:
$b=-7\to-D=23$
$\color{blue}{b=-6\to-D=2187=3×27^2}$
$b=-5\to-D=3919$
...
$b=+5\to-D=2759$
$\color{blue}{b=+6\to-D=1323=3×21^2}$
Solving the equation via Cardano's method for $b=+6$ gives the root
$-4-\sqrt[3]{49}-\sqrt[3]{56}=-(4+2\sqrt[3]{7}+\sqrt[3]{7^2}),$
where $56=2^3×7$ and $49=7^2$ imply that the root is in $\mathbb Q[\sqrt[3]{7}]$. The above root us the reciprocal os $\sqrt[3]7){7}-2$ treated above. A similar treatment for the $b=-6$ case gives a negative unit in $\mathbb Q[\sqrt[3]{3}]$:
$-(4+3\sqrt[3]{3}+2\sqrt[3]{3^2}).$
Neither field admits a solution for positive integral $a<12$, so both units given above are fundamental. While this example seems straightforward, most values of $m$ require much larger polynomial coefficients $a$ and $|b|$ to identify solutions in this way.