Is it OK to just hash a symmetric key to derive an authentication key?
Yes this is "OK".


  Please correct me if I'm wrong there.


You are correct.

It's effectively ratcheting the original key.

However there may be a more elegant solution such as using static keys for auth and ephemeral keys for key agreement (i.e. use Ephemeral-Ephemeral Diffie-Hellman for each new message).