Will $p$ always be prime if $p^p+(p-1)!$ is prime?

Question

While finding primes of the form $p^p+(p-1)!$ on PARI/GP, I noticed that $p$ is always prime if $p^p+(p-1)! \gt 2$ is prime. The search range was $p \le 10^5$.

Here are the solutions for $p\in\Bbb{+Z}$ for which $p^p+(p-1)!$ is prime that I got on PARI/GP:

1
2
3
11
43

Questions:

$(1)$ Will $p$ always be prime if $p^p+(p-1)! \gt 2$ is prime?

$(2)$ Are there finite primes of the form $p^p+(p-1)!$, where $p\in\Bbb{+Z}$ ? How would you prove/disprove this?

Edit: Just realized that the answer for the 1st question was obvious. But I think the second question will be much harder to answer.

Answer 1

If $p$ is composite, then it is divisible by some prime $q<p-1$. That $q$ obviously divides both $p^p$ and $(p-1)!$

Answer 2

$$7901^{7901}+7900!$$ is probable prime

http://factordb.com/index.php?id=1100000001296185249

I found it with pfgw