And that did it.

After my last reply, I removed the gamepad physically, and put the alan-1 flight yoke adapter in position one of the RPI.

Then, I deleted the Mame-2003-plus config files at \home\pi\RetroPie\roms\arcade\mame2003-plus\cfg

I went into Retroarch config from Retropie and went to bind inputs for port 1. I changed it from Gamepad to Gamepad with Analog, set it to left analog, then mapped only the analog and four buttons.

Next, started up Star Wars, and mapped input for this game to the Analog X/Y and buttons.

Deadzone gone!

Settings and everything held after reboot.

Thanks for all the help!