I have a hp7500a. The exact same problem. The advantage of this printer is that there is a window built in to the bottom of the paper tray very close to the problem. A rectangular hole with rounded edges, to be exact. My gear was just off the splines on the metal shaft. An easy fix with a good flashlight and a long flat screwdriver to slide the gear up and down.
I tried the above solution, and it worked, but I want to warn you of a Murphy's Law that caught me. I had stood the printer up on one side to apply the crazy glue. I applied (unknowingly, because I couldn't see what I was doing) more than one drop, and the crazy glue ran down the shaft and glued the pivot point where the roller "landing gears" allow the rollers to drop down. I had glued plastic to plastic, which usually isn't a big deal. It actually turned out to be better glued than I thought.
But , after the fix, I didn't know that the mechanism was glued in place. So now, after trying to print, I saw the rollers moving where they didn't before the fix, but the roller landing gear stayed in place, and didn't drop down to the paper!!
After much screwing around, I finally took a drop of 3in1 oil, placed it on where the crazy glue had set on the pivot, and after some time and pushing, freed up the "landing gear" for the rollers.
Now, everything is freed up, but they still won't lower. I probably broke some other gear while it was stuck in place.
The moral of the story is: 1. flip the printer over so it's lying on it's top (I took off the cover to the flat bed first). That way if there's too much crazy glue, it won't run and glue up something else. 2. Use a tiny amount of the crazy glue. Or use an epoxy that doesn't run like crazy glue.
BTW, I decided to enlarge the "window". I used a drill bit that's used to make large circular cuts in a door for a door handle. I had used a dremel wheel, but it spins too fast and melted the plastic tray.
One more thing, the plastic used for the body of the printer dissolves in acetone. I was going to try to dissolve the crazy glue with it, but I decided to test it first on the cut out window piece. Did something right!