I know Håstad's broadcast attack when e = 3, but what if e = getPrime(randint(350))?
If you have $e$ ciphertexts for the same message, then the attack is the same, you just have to apply the CRT with $e$ values, then computing the $e$'th root of the resulting value might need some work. And this is assuming that all moduli are relatively prime of course. If that's not the case, there is $i,j$ and $gcd(N_i, N_j) \neq 1$. So you can factor one of $N_i$ or $N_j$ and decrypt the message. Lastly, depending on the size of $e$ and that of the $N_i$ you might want to investigate attacks on "small private exponents" such as Wiener's attack or from Boneh et al.