Inverter Bad?

Thought i did that. I have newer house batteries, Trojans, I keep it plugged in when home.. I keep the house batteries on when home.  Heat kicks in at 45°, electric start,  but runs on propane, I rarely use the heat pumps in the AC system, (Electric). I unplug my fridge when home, so less draw.                          My inverter as I said is like a Christmas tree, plugged in or on the road, I took it to 3 NFL games the last 4 weeks. I tried the  stock, built in microwave and it must have overdrew the system. The inverter blinked red in the charging portion. I shut it down, reset it, and it  turned yellow then green in the charge portion. I could use the coffee cup, and toaster, but not the microwave. 

Hope this clears it up for you

Thanks to all of you! I will check out the batteries tomorrow. Can't imagine I got 4 new Trojans late last Spring and they are the culprits! Anything can happen!

I would start by charging and testing the batteries

Your post is super confusing. Not surprising if you don’t know much about RV electrical. 
separate the issues. Make sure they’re repeatable and isolated to inverter operation. 
Discern why or what you are trying to say about starting vs house power. 

You need a lot more diagnostics before you can determine where then problem is. It could be the inverter, batteries, converter, microwave, a relay, fuse, breaker, lose wire and probably a half a dozen other things. You need to start checking inputs and outputs on all the pertinent devices before you start throwing your money down a rabbit hole. One thing I can tell you, starting the engine of the RV with the battery bank at 10.5 Volts is more than likely not going to be sufficient to run the microwave. Secondly, it is highly unlikely that the stock systems in a 20 yr old RV will ever run the microwave without serious after market improvements. Most importantly, lacking a full and complete description of your whole system and I/O’s, no one on here could reliably diagnose your problem(s).

 what type are the house batteries, and how many? it almost sounds like the batteries are done, did you run the batteries dead at all?  it could be the inverter though.  when you say the volts drop to 10.5 to 11 V is that measuring at the actual batteries?  there shouldnt be that much of a drop if the batteries are good.  this one will take some trouble shooting.