I just went through this last weekend on my 93 F350 XL, crew cab, 4X4, automatic. My tach died also when I turned on the headlights. Also, my tach would jump in sync with the blinkers on and headlights off and had just installed a new light switch I did not have any burned wires, though. The windshield wipers were very intermittent and sluggish even after installing a brand new wiper motor under the hood and a new multifunction switch on the steering column.
The problem was a bad ground.
I pulled the plastic kick panel off of the passenger side between the front door and the glove box (end of dash). Behind that panel, I found three wiring harnesses there from the factory. On my truck these are for the roof cab cargo lights, the AC, and the windshield wiper circuit. Just for trouble-shooting purposes, I cut a piece of 12 guage copper wire about 10 feet long and went directly from the negative battery post passenger side to inside the cab pass side. I touched each of the black wires on all three harnesses and when I connected with the windshield wiper circuit all of the problems went away immediately. The tach came alive, no more needle bounce with the blinkers, and the windshield wipers work perfectly. The headlights and markers work perfectly. Even the little chime buzzer when the door is open with the key on came to life, and I haven´t heard that thing in years.
As a new project, I am going to find all of the grounding points on my truck and get them wired to battery negative, both sides driver and pass because my cab, fenders, and front rad support are rusted, and I guess the additional grounding can´t hurt anything, only help as the truck is going on 22 years old. By the way, I do not regret having purchased many new parts first and then finding the culprit to be a grounding issue afterwards. My truck is a keeper. Would replaced them at some point anyways. Hope this helps.
Russell