Re: trailer running light blowing fuse for revese lights
Based on your late-breaking news, you should determine the current needed to actuate the solenoid, add it to the current needed to operate your vehicle's backup lights, and compare the sum to the value of the fuse in the circuit. (The sum apparently exceeds the fuse value...) The solution is not to increase the fuse value. The solution is to install a relay actuated by the backup lights, but switching a sufficient 12V source to the solenoid.
Determine the solenoid current by research or measurement with a 12V source and an ammeter.
Relay can be a common horn relay, $4 from the auto parts store.
Don't know your available sources of 12V, but it's not going to be one of the other light circuits.
This is all assumes your solenoid is actually working correctly, is properly grounded, has good connections to it, and is not physically stuck.
Start there.
Someone could have been sneaky and put higher-wattage bulbs in your backup lights, too. (Or auxiliary ones, too.) Worth a check.
There will be other solutions offered, of course. But fixing it right once is always the best course of action.
Good luck!