What vulnerabilities would open up if PGP authentication didn't have integrity checking? It is my understanding that PGP has both an integrity check and an authentication check. What vulnerabilities would open up if it didn't have that integrity check?

It seems like the authentication check is also basically an integrity check because the authentication would be a signature of the message. So if the message changed, the signature would be off.
Digital signatures provide authentication, data integrity and non-repudiation. Thus, you are right to say that the authentication check is also basically an integrity check. If it didn't have an integrity check (i.e. no digital signatures) then you cannot be sure that the message you received is the original and unmodified version sent by the claimed sender. But I think in this case, since the authentication is also provided by the signatures, if there is no integrity then you wont have authentication too.