Looks like where Winnebago connected the actual headlight wiring to the Ford supplied wires.
Drawings do confirm that:
2003 Adventurer Headlight Wiring Diagram
Ford/Adventurer wiring image attached. Colors match your pictures.
You have the headlights working, now you'll notice how bad they are. After you aim them correctly, you should look at adding relays to get more voltage to the bulbs, it'll help a bunch.
Wait a minute, rereading your post. And now I also see the burnt connector. The pigtail you found on Amazon should work. hopefully that fixes things for you.
(And yes, Winnebago used cheap knock off '02-'05 Impala lights on the Adventurers in '03-'05 (I think, without looking if it was '06 things changed) Good, OEM, light would be nearly impossible to find these days, but plenty of other knockoffs to at least have clear plastic, if that's an issue.)
(On a whim, I actually checked some GM info, you can still buy the headlights new from GM. '02-'04 version anyway, '05 version is discontinued. The glare caps changed in the lights, just have to match the bulb type to whether there's a cap in it or not. Price is around $200 each, actually not that terrible for proper OEM lights.)