Hit the bench again and I think ya'll were right that my wrist was the problem. It wasn't an obvious amount, but I ever so slightly straightened my wrist to the point I thought the bar might slip out of my hands and the weight directed more into the flesh of my palm.
Another thing I did was make sure I displaced the weight evenly across my palms. My elbows go slightly outside my hand grip going down, and going up I have a tendency to apply pressure to the outside of my hands because of that. Now I concentrate on pushing the weight through the entire width of my palms. How much it helps I don't know, but it can't hurt.
So now I'm benching with no sharp pain in my hands. Thanks, everybody, for the help!
Glad to hear your problem is better. Keep an eye on it, speak up if it recurs.