You may have two problems , but often solving one fixes the other.
But if were me I would go after the Battery drain problem first.
Here is how I would go about it without a wire diagram.
Tools Needed :
1. Test Light ( with incandescent type bulb , not an L.E.D.)
safest way for someone that has never done this before .
2. optional an Ammeter ------ask me how to use if you do not know
Test Light method:
Disconnect the negative battery cable
Hook up the alligator clip end to the negative cable of bike
Use another alligator jumper wire and hook it up to the pointy part of test light.
Hook the other end of the jumper wire to the battery.
Now you essentially just put the test light in series.
With the key off , if you have a draw on the system the light will be Lit.
Now if Lit , first disconnect the trailer plug . If light goes out then your problem is in trailer wiring. (but it could be bike side)
If light stays Lit , go to your fuse box , pull each fuse one at a time until the light goes out.
Sometimes the light will go dimmer after pulling a fuse .
Keep track of this if this happens. But continue on until light goes completely out.
This will tell you the exact circuit(s) the problem is in .
Then troubleshoot that circuit or circuits.
p.s. you may have more than one fuse box on the bike
you may have inline fuses as well
if all else fails , you have disconnected each fuse one at a time , and the test light is still lit then disconnect the alternator or voltage regulator ( if separate of alternator).
You might try disconnect starter solenoid too if have one.
Alternators can go bad in such a way , that it will draw on a system , and no fuse to pull.
Unfortunatley I am not familiar with your bike I have a GL1800
But I do have a Valkyrie with the 1500 but less than 10,000 miles and never needed to troubleshoot yet.
You know, windows didn't get that bad over night! It took fifteen years of careful development!