Can I eliminate the expensive coax "multiple outlet" plug?

Ah. On cable, you don't need the antenna amplifier. Many amplifiers are designed to switch between cable and antenna when the amplifier is turned on and off, from what I understand. Assuming you get adequate reception with it off, no further work is needed. If you don't, verify that directly connecting the television to the park cable connection gives proper reception before attempting to troubleshoot the system in your RV; I'd think there would be a good chance that the difficulties lie in the park distribution system, and no amount of futzing with your RV will help that.

Try hooking the bedroom to a known good output. If it is the switch, you could put a splitter in another output, like the oft used VCR ;)

I have batteries and the amplifier works if I'm using the antenna but not if I'm using cable.

Where is your 12V power coming from? Is it fuzzy only when running with the converter on--or do you not have a battery at all?

It may well be that the converter is putting out nasty 12V power that is introducing noise when the amplifier is on. If you don't have a battery, there's a very very good chance of that.

If your factory/dealer isn't far away, that's where I'd be taking it.
Otherwise track every cable, make sure its wired right.
Also check.....make sure the antenna is being powered by the amplifier, which should be powered by 12v battery circuit

When I turn on the amplifier then the screen goes completely fuzz. And all this is how it came wired from the factory.

Yes you can do that, but you probably shouldn't. They do work, but you probably have an error in the cables.

1)make sure your amplifier is turned on.
2)try wiring direct from the output of the amp plate to the TV. Amazing how many times the cables are not even connected to the TV somewhere along the route.
3) check cable connections everywhere you can find them.