No Running Lights
I had the same problem with my '99 Suburban/'64 Overlander combination. Lost track of the number of running light fuses that were blown before discovering the problem. The last resort was disassembly of Bargman connector on the Suburban -- corrossion had briged between terminals on the connector and was causing the fuse to blow shortly after the trailer was connected. In my case, replacing the Bargman connector on the Suburban solved the problem (I now carry a spare for the car end of the connector as this problem has repeated about every 50,000 miles -- just about time to replace the current connector as the Suburban is up to just over 153,000 miles).
Good luck in finding the culprit of your blown fuses!
Kevin D. Allen
WBCCI (Lifetime Member)/VAC/Free Wheelers #6359
1964 Overlander International/1999 GMC K2500 Suburban (7400 VORTEC/4.11 Differentials)
1978 Argosy Minuet 6.0 Metre/1975 Cadillac Eldorado Convertible (8.2 Liter V8/2.70 Final Drive)