yer treatin' the factory b/u wiring like HD....it ain't
Do like one of the other posts said.
Get a relay that'll easily carry what the bigger lites draw
(the relay will be rated by AMPS and the lites will be rated by WATTS--- for every 100 WATTS of demand you need to be able to deliver about 9 AMPS of current. )
The relay will have, depending on design, either 3 or 4 leads.
Regardless of design, there will be a Heavy-in and a Heavy-out. You need a dedicated length of 10G copper (just because Dodge didn't run one back there) line from the battery (fused at about 20A, for safety) all the way to the rear work area. Terminate the battery-line at the Heavy-in. Run your Heavy-out lines over to your big bulbs.
Now, you're ready to wire the "trip" portion of the relay. Just run a tap-line (parallel splice) from the "hot" side of one factory backup light wiring down to the 'switch' terminal of the relay. If your relay has only the 3 terminals, you're done, except for a test.
The test consists, simply, of "ignition ON" (no need to start engine at all) -----gear-shift lever into "REVERSE"
If you've wired it all correct, your factory b/u lamps will light, just like always, but in addition, you'll hear a "click" or "clunk" caused by the tripping of the relay, and your new big bulbs will lite up, too.
If there's a 4th terminal on your relay, it SHOULD be a "ground", and can be routed right under its own mounting screw. If it's got only 3 terminals, its PROBABLY self-grounding.
Ground the new bulbs return wire directly to their own mounting screws.
As for the theory that you have a grounding problem, no. You have an overload problem.
For the relay, the one guy used a 30-A one... . I, being the overkill nut that I am, used a 4-terminal solenoid that was intended for a Ford starter. (makes a big "audible" when it engages and disengages)

I mounted it behind and above the LR fenderwell where it would be out of harm's way, unless I go somewhere and totally submerge the rear axle. .