consequences of using fixed values with OAEP
If you encrypt something with low entropy but use a fixed seed for OAEP, it is trivial to brute force it, and verify your guesses with public key, while randomized all or nothing padding will make verifying a guess impossible. 

In the context of signatures rather than encryption, it doesn't seem as severe, but you are supplying the attacker with a bunch of related plain text cipher text pairs which can't be a good idea.