Trigger shifters partly malfunctioning I cleaned my wife's 1999-vintage mountain bike, and because no good deed goes unpunished, I seem to have caused a problem in the process.

This bike has a Shimano Deore 9-speed drivetrain with trigger shifters. Starting from the smallest cog, the trigger shifter for the rear derailleur can downshift through the first three gears correctly; after that, it needs to rotate back for the ratcheting mechanism to reset. That's where it goes wrong: the pawl apparently won't engage with the ratchet wheel, and the trigger swings freely.

I've been brave enough to remove the housing over the mechanism and blow some compressed air in there, which did not fix the problem. There are enough small parts in there that I'm reluctant to go further without some guidance (if it's possible for a home mechanic to fix at all).

I see that I can apparently buy a replacement for about $24, so if the correct answer is "replace it" I won't be too broken up.
Get yourself some Clean Streak degreaser and a dental pick. Gently work the pawls until they snap back into place firmly. Then use the compressed air to clean it out and put a drop of thin chain oil onto the pivots. That should get you going.