Since you said it doesn't matter if the trailer is hooked up or not, the problem is in the truck. In over 30 years of trailering, I found most problems are at the trailer plug receptacle, either the receptacle itself, or the wiring connecting it to the truck's wiring.
Start with the trailer plug receptacle. Look at the wiring inside to see if any of the wires are touching. Try disconnecting it and see if you still have a short. If it's an aftermarket receptacle, check the splice used to wire it into the truck's circuit. Some after market trailer plug receptacles plug into the stock harness. If this is your case, try unplugging it and see it the short still exists. If the short is gone, replace the whole unit.
Next place I'd check is the brake controller wiring.
If the short isn't in either of these two places, go back to the plug and start working your way back on the plug's wiring harness looking for anywhere there may be a pinch point.
For test equipment, you can use an ohm meter. Connecting the ohm meter to the hot side and ground will indicate a short if the meter reads "0" ohms (or a very low reading like 3 ohms, etc.).
Another method is to replace the fuse with a 12 volt light. If the light comes on, you still have a short. This method will save you fuses while testing but may require a second person to watch the light while you work.