The gun was pump-shooting, so you pump the gun and the water+air shoots right out the front. I decided to take a crack at opening the gun and fixing the problem.
The screws were really rusty. I ended up drilling out two of them.
Check out my Minus page for all of the screw locations.
The pump cap was easy, I just had to slip the spine (back edge) of my knife into the gap and slide it like a wedge until the thickness pushed the cap off. It would have been hard, but I had a lot of practice and the perfect tool. Then I wedged the shell open and removed the pump for better access to the nozzle cover.
The real problem came when I tried to get the nozzle cover off. Thankfully, it was not glued, but I did spend a good deal of time trying to get the stupid thing off without just breaking it. Luckily, the plastic was quite resilient and didn't chip or snap, only bend. I did stress out the plastic a lot more than I would have liked though.
I tried all of my techniques to get this one off. Normally I twist the shell from within the nozzle cap, this helps separate one half of one side of the shell from the nozzle cover. However, I soon realized that there were two screws underneath, so that wouldn't work. I finally pried the cap just enough with a flat screwdriver that I could get another screwdriver into this screw hole:

Then I pried.. and pop! One half snaps loose... and then a little more work around the edge... and presto!

Those two dammed screws... I know where they live now.

I see you...
Now the gun seemed to be open, except that right around the PC and firing valve, the shell was stuck together. I managed to wedge the thing apart and use pliers to remove the lever rod from one half of the shell.
Surprisingly, at this point the gun worked fine. When I put everything back together, the valve gets stuck open again. And that is when I realized the problem:

The rust on this rod is preventing the plastic bit from turning easily because of the friction from where the rod sits in the shell. I can see how people would just shove a spring or a rubber band or something into the gun to fix it temporarily, but I wanted a long term fix. I wanted to use some WD-40 and remove the rust, and then smooth out the rod (and possibly the holes in the case) so that everything moves smoothly.
Here is the lever rod again (in the center) and the screwdriver points out the hole in the other half of the shell.

I couldn't find any WD-40, but I did randomly find a 9/32" drill bit lying around. By some coincidence, it was exactly the right size for widening the hole! I drilled into the hole and extracted a few shavings of yellow plastic. I drilled both sides, wiggling the drill around just enough to make the whole hole wider by just a fraction of a fraction. I closed it up... and bam! Fixed. Now the trigger opens and closes the valve every time.
At this point I took a razor blade and trimmed most of the plastic around the inside edge of the nozzle cover, in case I ever have to open it again. I popped the cover back on, and even though I removed so much of that plastic ridge, there is still no way the nozzle cover is going to come off easily. Same goes with the pump cap, I had to hammer that sucker back on. Here is the end result:

Good as new. Or at least, as good as moderately used but still working like new. And no rubber bands required!