Welcome aboard!
1. Unless you have a residential refrigerator you likely don't have an inverter.
2. You have a converter which converts the incoming 120v to 12v to operate lights, fridge, etc. and to also charge your batteries while connected
3. The AC runs only on 120v and will only work when plugged in. Your AC has nothing to do with your 12v system. Make sure the fan is set to Auto...not Low or High. That should keep it from running all the time.
4. Dimming lights:
---a. you could have bad batteries. How old are they? I would remove them and have them load-tested
---b. you could also have a bad converter. A properly working converter can run your 12v systems without batteries while you're plugged in to 30a/120v.
Last summer I had to replace both the converter and batteries at the same time.
There could be other things at play but checking the output of the converter and the quality of your batteries is a good start.
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Jeff & Sandi (and Teddy - 7lb Schnorkie)
2018 Montana High Country 305RL
2015 RAM 3500 Crew Cab 4x4 DRW
Demco Recon Hitch on RAM Puck Ball
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