My 2001 Ford Ranger 4R44E transmission won't shift into any gears except 1,2,3. Is there any way that I can fix it other than replacing the entire transmission, which I know will cost a pretty penny?

when an automatic box cannot engage a gear you know its gona be expensive..
as you might know , the automatic box has a lot of valves etc..the problem could just be a bad one.
go to a garage and get a price on it.

Try cleaning the valve body and change the trans filter also check the modulator valve ,it may be dirty.