2004 F150 Quad cab Lariat (newer body style). Low beam headlights won't turn off. Unplugged fuses to stop them from running battery down. New battery. Hi-beams and parking lights working normally. I checked relay box in engine compartment and it looks like one in owners manual that is for "Without Daytime Running Lamp (DRL) option. Unplugged R201 and it did not stop the lights. Is there some other relay in the system? Could this be the headlight switch?

SOURCE: 2004 ford focus headlights

You need to replace the high/low beam switch

SOURCE: 1997 Ford Explorer XLT: Both low beam headlights

Have you checked both low beam bulbs? I have seen both burn out close together.Does this have tilt steering wheel? If it does,remove the plastic steering column cover.Check the wiring to the dimmer/turn switch.The connector ends would pull out of the hard shell connector,when using the tilt feature.The wire harness was mounted too tight,and didn't allow for the angle in tilt.

SOURCE: 2001 Ford Focus - High beam / low beam problem with headlights

just replace the dimmer switch under the column its wore out on the low beam side

Testimonial: "It's great to know that there's not a whole lot more to check! I wouldn't have guessed I was so close. Thanks!"

SOURCE: 98 ford lariat. Headlights stopped

Is this both the high beams and low beams that do not work ?

If you checked all 3 fuses with a meter ( with the headlamp in Low position for both low beam fuses or in high for the high beam fuse ), the next items to test are the MFS ( MultiFunction Switch ) and the main headlamp switch ( test not look at it ).

The main headlamp test chart


The test chart for the MFS

SOURCE: 2005 Ford Focus ZX3 SES, the blue high beam

It sounds like you are confused about how your headlamp system is supposed to work. The DRL does not run on the High beam circuit, it runs on the low beam circuit. The fog lamps are supposed operate with the park lamps and also with the low beams, not the high beams. From your description, I would say that your system is operating as designed and you are having trouble telling High from Low. (Don't feel bad, that is why high beam indicators were invented in the first place.)