File this info under "So now you tell me!"
LED turn signals require an LED-compatible flasher. The one in the coach likely is not LED-compatible; the one in the Jeep may be compatible.
If you replace the bulbs with LEDs and get "hyperflashing" (very rapid flashing), then you need to replace the flasher.
Here's a link
to an LED-compatible flasher I purchased and installed on our former 1997 coach, built on a Ford E-450 SuperDuty chassis.
I elected to stay with the bulbs on our Honda Fit toad, because the flasher is part of the instrument cluster, and therefore difficult to replace. Therefore, on our former rig the coach and toad bulbs- tied together through the umbilical and wiring- were a mix of LED (coach) and incandescent (toad). This worked fine because the LED-compatible flasher in the coach can handle both types.
You may find that the LED replacements don't result in enough improvement, in which case the "black box" (or relay set) with its own 12V source is the next "fix."
The best way to test the whole thing is, at night, to connect coach and toad, turn on the coach tail lights, hit the hazard flashers, and then back up 100 to 150 feet. If you can't get enough visibility, you need something more than LED replacements.