I don't think bulbs are the problem, because the hazards are working. There are only two filaments in the rear bulbs. One filament is for running lights, the other filament is for the brake lights, turn signals and hazards. However, they are very cheap to replace, and worth a shot if all else fails.
I also doubt it's the flasher relay, because the hazards share it, and are working. It is very cheap to replace, and worth a shot if all else fails.
I looked at my owners manual, turns out that the fuses for the hazards is NOT the same for the turn signals. Try replacing Fuse 5 (or both 4 and 5 for good measure). EDIT: That's for a 94, check your owners manual in case they made a change.
I didn't see anything on the relay list (power distribution box under the hood) that looked appropriate, so you should be good there.
How did you "check" the multifunction switch? If you didn't put a voltmeter on it and try the contacts for the turn signals, I wouldn't rule out that PART of the multifunction switch is bad.
Now, having said that, my wife's car (not an Expolorer) had an issue with the turn signals. I had to remove the multifunction switch and clean the contacts. They used some grease during manufacturing which managed to collect dirt (or something) and short across the terminals. The "good" grease was clear, the "bad" grease was black. My point is, if the fuses don't fix it, pull the multifunction switch and look and see if it needs cleaning.
If all else fails, and before you bring it to a garage, one other thing to consider (and this is mostly theory): the hazards seem to over-rule the brake lights and turn signals. If for some reason, the hazard switch is not resetting correctly when you turn it off, it may not allow the turn signals to work correctly. Are you sure you have brake lights that are working? I would pull all the steering column covers and see what might need to be cleaned, check the wires for shorts, and maybe replace the hazard switch for good measure.
Hope this helps, let us know what happens.
Mike
