The white wires are the grounds for each bulb. You say it works with them not connected to anything, but I have no idea how that could be. In the original wiring, the ground connection for each fixture may be provided by the screw that attaches the fixture to the aluminum skin. In your pictures, I see no connection between the shell of the bulb and the screws that pass through the fixture.
For example, in your lower left picture, I see two bulbs. Look at the one on the left. It has a red wire going into the socket, but nothing coming out. In order to light up, the metal shell of the socket must be connected in some way to ground - that is, to the metal parts of the trailer. Specifically the aluminum skin. I would bet that in the original fixture, one of the metal mounting screws passes through a metal plate of some kind, which in turn is part of the socket. The screw makes electrical contact between the metal plate and the aluminum skin of the trailer. In that same picture I see four holes in the corners of the white plastic base plate, but there is no metal plate going to one of the screw holes. Instead, the ground connection is made via the white wire on the back of the socket. In other words, you must find some way to make the white wire contact the aluminum skin of the trailer.
I think the pictures you are showing us are pictures of the new fixtures. Can you take the lens off one of the old fixtures, and show us a picture of that fixture?
I wouldn't rip those white wires off until you figure this out.
Bill
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