An accurate diagnosis of why the OP's RV A/C isn't working right using the household receptacle is going to be impossible from a keyboard. There are just too many things in the chain to pin it to one part being the culprit.
If the OP could budget for it, installing an RV receptacle, the necessary wire, and load center upgrade is going to be the least problematic way to power the RV when at home.
FWIW, I bit the proverbial bullet and installed a 50 amp receptacle in my garage while I was restoring my trailer. Because I had already installed a subpanel for a welder, it was not that much work. I added the breaker and pulled new wire from the load center to the receptacle. The downside for me is that I presently have just a 25' shore cord. I will make or buy another 50 amp extension cord when motivated.
