Rear derailer shifts fully only in middle of cassette and loses alignment with cogs over time
There's not much that could happen to cause the shifter to pull and hold an incorrect amount of cable when you click it. 

A simple test to see what the shifter is doing is to loosen the cable anchor (or slacken the system by pulling a housing out of a stop), pull the housing away from the shifter, pull the exposed cable there gently taut with pliers, and shift back and fourth through the range. You'll be able to see if it's working normally, missing shifts, releasing too many steps at once, etc.

Once they're this old, Rapidfire/STI shifters are prone to issues with lubrication around the pawls getting gummed up, causing engagement problems. From your description it sounds like this may be what's going on. This is common and easy to fix with some cleaning and relubrication. However, it's also the primary mechanical problem with these shifters that's fixable at all. If there's a broken spring involved, which is also pretty common, there's not much you can do.

To service them, the important thing is to never under any circumstances take apart any part of the mechanism itself. Depending on the shifter, there are usually covers that can be removed or nudged aside to let you spray the guts and/or work the pawls around. The usually recommendation is to use a silicone spray lube because it won't degrade plastic. If possible I like to put a small dot of grease at the pawls, roughly emulating how they come new.