I'll try to help.
I agree, don't hook them up to your high beams. In fog-snow highs bounce glare back at you. An only key on connection would work--- one the low amp side of the relay. But don't use a fuse that would shut your engine down. Add a fuse on the hot side of a fuse, I soldered an "add a fuse" on a fuse to run my intercepter gauge. Hooked up in the box by the drivers door.
Use a unused large fuse from the mail box under the hood for your main draw, I hook uped my trailer battery there.
Could you scan the relay instructions ? I don't know how this relay is made--sounds like the switch is on the hot side, not path to ground.
So many wires, where do they all go?
Some hot Some grounded.

PS I put in "Truview Capsule " head lights from Napa-- Less glare I like them.
2002 GMC 1/2 ton Sierra,4WD, Xct cab, camed, mid-length headers, Nelson tune, electric rad. fan, trany cooler, Heavy 5750 lbs with topper.