Backup lights usually come on anytime the ignintion key is in RUN and the shifter's in R. We have an OP who added a switch to turn them on anytime to provide light to work on the towbar, clever idea.
I'm not about to say This is the Problem This Time, but I can promise you that Loose or Weak Grounds can cause all manner of Gremlins. Try making up a couple temporary grounds. Backup Lamp Bulb Shell to Known Good Ground, and Marker Light Lamp Shell to Known Good Ground. Known Good, like the Chassis.
If God's Your Co-Pilot Move Over, jd
2003 Jayco Escapade 31A on 2002 Ford E450 V10 4R100 218" WB