In a group of order 21,
The number of 3-sylow groups is equal to 1 mod 3 and divides $7$, so the possibilities are $1$ and $7$.
The number of 7-sylow groups is equal to 1 mod 7 and divides 3, so the possibilities is only $1$, there is a unique and hence normal 7-sylow group.
You have already dealt with the case of a unique 3-sylow group so let's consider if there were $7\quad 3$-sylow groups:
A 3-sylow group in this case is just $C_3$ we can see if any two 3-sylow groups overlap (except the identity) they are equal so we have 14+1 (the identity) elements from 3-sylow groups leaving $6$ elements over exactly what we need for the 7-sylow $C_7$. So it seems like at least one group like this will exist.
Now you can try to find a way to construct it but not using Sylow theory.