scotch
+1y
You need to track down that short. Whenever somebody comes on and says that their running/parking/dash lights are blowing the fuse, 8 out of 10 times, it's because a "radio illumination" wire has been cut and left dangling behind the dash when a previous owner upgraded the radio and saw no need for that wire. It wasn't shorting anything out during the install because the lights weren't on. And maybe it didn't short for a long time but has since rattled loose and is touching the transmission tunnel or the dash framework. So that is something I would definitely check out. The other common issue is if somebody has tapped into the existing lights to run fog lights or something like that, and left some wires less than secure. I would closely inspect the wiring going to the parking lights. And as has been mentioned, to the rear for the trailer lights.
If you start the truck and then disconnect the battery, what voltage are you getting then? Your battery alone should be putting out the voltage you are getting now. If you disconnect the battery and then measure the system voltage (on the harness terminals), you'll be reading the alternator voltage without the battery being connected. If it goes up to the 13-14 volt range, then you'll know it was the battery dragging the alternator down. If you still can't get more than 13 volts (system/alternator) with the battery disconnected, then your alternator is weak. Pull as many fuses as you can but still have the truck running (battery still disconnected) and see if you can't get any more out of it. If it goes up significantly with fuses pulled, then it is something else dragging the system voltage down.
Charge the battery and take it down to the auto parts store and have them load test it. I have seen bad batteries as well.
Find out exactly what relay is buzzing. Get a picture of it and post it if you can. Somebody here will recognize it.