So here is what happened in the end. I moved, and the PC in my arcade didn't survive. I bought a replacement with Windows 10 in it, and it wouldn't read both joysticks. I thought for sure it was Win10 assigning a single input device for the older program, so I was trying to figure my way around that. In the end I couldn't, so I was going to upgrade my mame although I was really happy with it and didn't want to mess up the good thing that I had.
In the Ultramap software for my u360s, I was set as joy 1 and 2. In windows, I had 2 joysticks connected. In Mame, I was configured for 1 and 2. For some reason though, joystick 1 was read by mame as player 3, and joystick 2 was player 1, even though joystick 1 was working as player 1 and joystick 2 wasn't doing anything.
Nonsense!
So in Ultramap I set joystick 2 as Joystick 1, and 1 as 3. Does that make sense!?!?!??? No!?!?!?!??? But it's working now.