Mt 03 GT started blowing the fuse several months ago every time I put it in reverse. It was "beep", blow. The switch for the beep and reverse lights is on the transmission so I just unplugged it. I'm guessing either the switch is bad or there is a short from the switch to the lights, but I haven't bothered with it. I'll probably just jump the harness and see if it blows again. That would let me know if the switch is bad or the wiring. I just use my brake lights when backing up for now.
Just thought of something since you said if you move the shifter it started to work. I wonder what inside the transmission activates the switch. Hopefully it is not something that is broken or loose on yours.
Thomas
the "Mustang Guy"
1987 5.0 LX Mustang
2016 Mustang GT - current daily
2004 Satellite Vibe &
2009 Red Vibe GT -twin's cars
2003 Neptune Vibe GT - prior daily
2010 Red Vibe GT - RIP 6/16/14
2006 Platinum Vibe - son's car