Since you're using a normal grip for your presses, the first thing I'd do is to make an honest assessment of how much your wrist is rolling back (toward your shoulder) when you're doing the presses. Ideally have someone knowledgeable watch you bench or take a vid.
The bar should be more or less directly over your forearm at all times when pressing. Many people with wrist pain of the sort you're talking about let their wrist roll back, or put another way place the bar too far up in their hand, so that the forearm muscles/tendons have to support the weight. Often times people may not even do it to the same degree in each hand and have no real idea that they are doing it, unless looking for it specifically.
Another thing you could try, regardless of whether you're doing the above, is pressing with a false/thumbless grip. Doing so often eliminates all wrist pain for people when pressing.