Many thanks to Noam Elkies for taking the time and effort to resolve this question and for sending me the results by email.

His, summarized, answer is:

The smallest primes for $n=11$ and $n=12$ are respectively $p=1057543811051633$ and $p=1448734752622601$.

For $(n, p) = (12, 1448734752622601)$. There are no other examples of $n>11$ for prime $p$ less than $7.5 * 10^{15}$.

Many thanks to Noam Elkies for taking the time and effort to resolve the first part of this question and for sending me the results by email.

His, summarized, answer is:

The smallest primes for $n=11$ and $n=12$ are respectively $p=1057543811051633$ and $p=1448734752622601$.

For $(n, p) = (12, 1448734752622601)$. There are no other examples of $n>11$ for prime $p$ less than $7.5 * 10^{15}$.