How do you fix the back doors on a 1989 Lincoln Town Car that will not open from the outside while it is unlocked?

Sounds like you need to replace the outside door handles. There is a piviot section that wears &/or breaks stopping the outside handle catching the locking mechanism. It is a bit fiddly, but after the first one they become easier. I have had to replace both rear handles on my 84 & 88 town cars - both use the same handle. Also check to see if the child safety lever has been enacted,it is located near the latch on the door.

Aussie Bill