I have significant back and forth wiggle in the freewheel of my '88 GT mountain bike. Could this cause the rear deraillieur not to shift properly? Do I need a new freewheel, or is there an adjustment? If a new part is required, what all is involved in changing it?

Was this wiggle there before you started having shifting problems? Is the wiggle only noticable while coasting? If so this is normal and just ignore it. If the side to side wiggle occurs while pedaling and your having difficulty adjusting the derailleur then you could have a freewheel cross threaded onto the hub: did you replace the rear hub or have a new wheel built up? If it's crossthreaded then your probably going to need a new hub and freewheel. There are no adjustments possible on a freewheel.

To show how to fix that would be a pain to describe here and the typing involved so go here instead, plus you get pics: https://www.parktool.com/repair/byreg...mageField2.y=5