U1 works fine. But When U2 gets to where I can control the character my mouse is bouncing and jumping around all over the right side of my screen. If I drag it left then it just jumps back right. I thought it was my joystick interfering with it like usual but it's not even plugged in.

I've had the same problem. When I first read your post, I thought it was your joystick as well. Apart from disconnecting it, you might want to open up the file: 'dosboxULTIMA2.conf'. This contains a section called [joystick]. It should read this:

joysticktype=none
timed=true
autofire=false
swap34=false
buttonwrap=true

Doing this should eliminate all possible unwanted input. Be it generated by the joystick itself or some other source.

I have seen this problem when alt tabbing. Once back in the game I push tab again and it stops.

It may have been the alt tabbing. I do that a lot. Next time in game it didn't do it.

I can confirm that this is a result of alt-tabbing DosBox.

It doesn't happen every time though.

It's the tab key that does it. When you alt-tab, the game might end up thinking the tab key is still pressed when it switches, so when you go back to the game, it's acting like tab is being pressed down. Tab is used as a key in the game for cursor movement, so that's what it's doing.

If you hold down the tab key in game, you'll see it do it as if you alt-tabbed. To make it stop doing it, just press the tab key, as sollus suggested.