There is an ultrastick plugin on the GameEx forums I believe. Maybe you need to use that.
I'm familiar that the plugin exists, but, if I read it correctly, it's really designed to download the appropriate map for the appropriate games. I don't think it solves getting the right control working on the right games, something I think Mame is responsible for.
Interestingly, I don't know if I'm going to use the plugin, though it would make downloading maps to the joysticks easier. There's WAY too many restrictions and pathing issues. I don't want to rearrange my hard drive and reset all my other INI's just to use this plugin. Of course, I haven't even started to configure the per game maps yet. I may not have another option.
Howard_Casto helped me solve my problem with getting the controls to work properly.
The joysticks are digital. Something I knew, but it didn't occur to me that mame required tweaking for them. So, I guess my setup is a special case where my mame.ini has to have this:
#
#INPUT DEVICE OPTIONS
#
joystick 1
mouse 0
And, because I have a Happ Trackball with USB conversion and a USB Slickstick Tornado spinner, the following section of my mame.ini has to be set this way:
#
#AUTOMATIC DEVICE SELECTION OPTIONS
#
paddle_device mouse
dial_device mouse
trackball_device mouse
I tested it pretty thoroughly yesterday, and with the exception of a few games, the joys don't interfere with gameplay on trackball and spinner games, and visa versa. In fact, yesterday was the first time that I was able to use ONLY my front end to select and play games. My feet are sore from standing in front of my cabinet and playing all night.

I had such a hard time finding help with this one. I must have spent 4 hours searching for a solution, as well as trial and error testing. With that, I'm gonna rename this topic to hopefully make it easier to find through Google or the search functionality here. It might be wise to put this in the FAQ (or somewhere) over at Tom Speirs' place. I can't possibly be the only person who has experienced this problem.
Thanks Howard. You're suggestion worked well.