I had the same problem. Also, the dome light would come on going down the road. I found if i thumped the dash, it would go out.
Later I had trouble with the battery as you described.
Turned out it was the instrument cluster causing all the problems. The replacement has to be an exact PN match. Mine was a 7L34-10849-AAB. The first cluster I purchased used off ebay was supposed to be compatable with the PN I had but when installed, my power windows, radio and auto headlamps quit working. I returned it and purchased another that had the correct PN in the photo. I got my windows and radio back but not the headlamps and the battery now stays up fine,

Two outa three ain't bad.
I never would have guessed that it did that much!

 

FYI.

About a year ago the parking lights on my 2004 Lariat started coming on by themselves overnight, draining the battery. Initial testing resulted in changing the ignition switch, to no avail. Further extensive testing identified BCM as cause. Discovered it has two relays inside, one for headlights and the other for turning on/off the other automatic lights. Replaced the one for headlights and ruined BCM. Considered replacing the BCM with either a new or used one, but decided the high cost was not worth it to me. Sent it for repair, but got charged without fixing it. Ended up disconnecting the BCM. Now I use the key to open the doors, and control the lights manually. Also re-wired driver's door master lock switch.

In retrospect, I would operate the lights manually. The remote control would still work then.

 

My issue of my headlights staying on even after truck turned off is now gone [or at least unless it comes back]. But apparently it was some kind of electrical gremlin lurking from a screwed up battery jump [*see note below]. I chased it down for few years changing the main light switch, turn signal switch, and ultimately drove with fuse 40 pulled out and reinserted at night if I needed headlights and pulled again when I was done for the night so headlights wouldn't drain battery overnight. Either that or would need to disconnect the battery. Then other day I decided to check every fuse/relay related to DRL and couple others. None were bad, yet the problem went away. So I'm thinking a charge was held somewhere and possibly released when I pulled the fuses, or went elsewhere, TBD. So these are the fuses and relay I checked. Maybe this will help someone else, and avoid wasting hundreds of dollars with a shop, or many hours of their own time.
14, 23,25, 37-40, R203

*I was stuck in a store parking lot with dead battery and a driving teenager with his father offered to assist and said he had a new "jumpbox" for this reason. So I let them handle it and next thing I see is a small wire caught fire near the terminal. He/they hooked it up backwards. How do you mismatch red & black to wrong cables??? Point of share is ALWAYS HANDLE YOUR OWN VEHICLE

 

Substitute a known good headlight switch before messing with the VSM
Grab a shop manual? Go through the flowchart,. about half way through the diagnostics the book will say to substitute a known good part
You have jumped through some hoops if you know the sunload sensor is turning the relay on
I sure would verify every bulb in the truck is the correct one for the truck, and no funny business with LED lamps right?
Ghost in the machine type stuff happens with LED lamps because the bulb resistance is not there

 

I stated I changed the headlight switch

 

Well, it might be no good

 

So you think a new switch is no good? AND the old switch is still good. Proven by fact the issue is resolved.
Did you not read my entire post? Nothing was defective. That was the point.

If you were trying to help i do appreciate that.

 

To say again...

My issue of my headlights staying on even after truck turned off is now GONE, no more, eradicated [or at least unless it comes back]. But apparently it was some kind of electrical gremlin lurking from a screwed up battery jump [*see note below]. I chased it down for few years changing the main light switch, turn signal switch, and ultimately drove with fuse 40 pulled out and reinserted at night if I needed headlights and pulled again when I was done for the night so headlights wouldn't drain battery overnight. Either that or would need to disconnect the battery. Then other day I decided to check every fuse/relay related to DRL and couple others. None were bad, yet the problem went away. So I'm thinking a charge was held somewhere and possibly released when I pulled the fuses, or went elsewhere, TBD. So these are the fuses and relay I checked. Maybe this will help someone else, and avoid wasting hundreds of dollars with a shop, or many hours of their own time.
14, 23,25, 37-40, R203

*I was stuck in a store parking lot with dead battery and a driving teenager with his father offered to assist and said he had a new "jumpbox" for this reason. So I let them handle it and next thing I see is a small wire caught fire near the terminal. He/they hooked it up backwards. How do you mismatch red & black to wrong cables??? Point of share is ALWAYS HANDLE YOUR OWN VEHICLE