To diagnose your problem you have to figure out what is powering the lights. Your headlamp system has uses three fuses for the headlamps, a 100 amp maxi-fuse (large size fuse) in the battery junction box, (the fuse box in the engine compartment) then two regular size 10amp fuses in the same battery junction box, for each headlight. I am guessing that the fuse you are removing to kill the lights is the 100 amp one. When you remove that fuse do you hear a click sound before the lights go out? If you do, you may have a short circuit to ground on circuit 292 a brown wire from the headlight switch to the headlamp relay or a bad headlight switch.. The 100 amp fuse powers both the main power for the headlights and the low current pull-in windings for the headlamp relay and those pull-in windings should only be energized when the headlamp switch is turned on. It is the headlamp switch that grounds the headlamp relay pull-in windings. If there is no click when the 100 amp fuse is removed, then you may have a stuck relay. To test the relay, (I would keep the engine running as not to run the battery down) put the fuse in so the headlights are on, then tap on the relays in the battery junction box, (it is best to use the handle of a screwdriver, hold the blade end of the screwdriver and use the handle as the hammer) you want to hit one relay at a time.
You are trying to jar the relay contacts apart not break the relay open, so you don't want to hit them too hard. If the relay contacts are stuck and hitting it releases them, then you should be able to replace the relay to cure your problem. If that does not cure your problem, you can disconnect the headlamp switch, if the lights go out, the switch is stuck in the on position. Lastly, if those steps did not turn off the lights, you may have a short to ground on the brown wire between the headlight switch and the battery junction box. Good luck.