This kind of "crosstalk" problem frequently stems from a poor or intermittent ground connection.
The Winnebago drawings indicate a single ground for all the rear and trailer hitch light signals. The wire is white, 14-gauge. Find it and check it's condition.
As the trailer hitch connector and coach lights share a common ground, this is where I suggest you open the hitch connector and ensure everything there is OK.
There is also the possibility that the backup camera trigger connection is poorly-made, shorting the other signals, Trace the source for that trigger signal and check its condition.
The positive news is that there's no common Ford connection for the turn, brake and clearance lights at the front of the coach. The signals do come together in the rear of the coach. Crawl under the coach and look for two Ford connectors inside the left frame rail, about 4 to 8 feet from the rear. One connector may be larger than the other. The wiring on these connectors may be as shown in this other post I made
; there are variations model year-to-model year, and your 2003 chassis may be wired differently than my 2007. In any case, find the connectors and check if they appear sound (don't take them apart or unwrap the loom unless you have to).
Finally, you may want to send a private message to iRV2 user "grtharris." Grant knows a lot about Fords and Winnebago Fords in particular.