If a light overhaul, cleaning and adjustment don't do the trick:
I have an 89" (or so ) GT with the same set up. Many many miles have passed and I have the same problems. The chain is shot, so is the biopace chainring and cassette. The two small cogs (can't recall the name) and bottom bracket need work too. Then comes the cables (easy) and next the shifters, I could try to clean them up but how much longer could they last? My shifters are not integrated, but I can't find replacements (too old), so I must go to grip shifters, or to integrated ones, which usually require an upgrade to v-brakes.
In my experience this was my excuse to buy a new bike. If you got the bread and need a ride that will last, get what you need. Much of the new stuff will blow you away, it did me.
If you are too in love to let it go consider buying complete used bikes fromt he goodwill and parting bikes together.
Hope this helps,
ehenz