[wiswind]

Try removing the bulbs from the rear, right side......maybe one of the filaments is shorted to something.......still lighting but pulling a lot of current.
If the rear bulb is lighting up....it is not likely a problem...but worth a try.......
The diagram for my '96 shows a single filament bulb used for the rear turn signals........with a double filament bulb used for the rear brake and side marker lamp.....1 filament for turn signal......another filament (in the same bulb) for the marker light.......but I do not know if this is true for your 2003.

I would also give the wires that you can find a real good look.......try the signal with the rear bulbs back in and the bulbs out of the front right.......does the "blinker" still go fast?
If the blinker is "going fast".....it usually means that there is extra current being drawn in the circuit.
Particularly, if it does it for only one direction, as the same blinker relay is used for both left and right (and hazard).
That is according to the drawing for my '96.

The front turn signal uses a dual filament bulb.....1 filament for the turn signal.....another filament for the parking lights.

If you have a conventional (not digital) dash display.....the signal indicator for each direction is also a light bulb.....which is another bulb that could have a shorted filament.......

The other question is if you have any trailer lighting connector.......if so......DIY wiring there is a possibility.....so I would look that over.

I have seen posts about a rear light module on newer years.....hopefully this is not a problem in your case.......try to eliminate the other "cheap and easy" solutions first.