I'm resonably sure the '87 used headlight relays to keep the switch cooler, but I'm not positive. If that's the case there may be a short to the relays or loose wire, I'm sure they would share a common power and ground.
I have the '96 and it has no relays, the switch gets really hot and the system sucks. Several guys over at
LincolnsOnline.com have installed relay sets in the 90's thru 97's to correct the fomoco flaw. Check over there if nothing else comes this way. I already did a search over there, is seems as if your problem is unique but you may want to ask those guys anyway.