30 AMP is AC. Most 38FT Class A have 2 AC units and are 50 amp. Is your RV a 50 amp RV or 30 amp? Which adapter are you using to reduce the RV power from 50 amp to 30 amp. Who installed your 30 amp plug in the man cave? Many have had the wrong outlet installed (even by electricians). A hot cord is a bad thing, either pulling to many amps, bad connections/prongs, or wrong outlet. People have done damage to their RV electrical systems with the wrong outlets, if that happened all of the below could be possible.
Depending on what else you had on, you could have tripped a breaker when the AC started up. Now the lights and TV question has 2 answers. Most TV's 120v AC and most lights in an RV are 12v DC.
Generally when a big load like the AC tries to start and there isn't enough power available it trips that breaker. You could have tripped the breaker for the converter( happened to be charging at the time), killing your battery, if lights or something else was left on. Check battery voltage.
Is your TV 120v AC or 12v DC? (house plug = 120v AC, big round plug =12v DC)
Did the breaker trip for your converter (loss of 12v DC?) check breaker panel, reset breaker.
You may have also tripped a GFCI somewhere in the RV that the TV is on the same circuit. Find all GFCI in RV test and reset them all. make sure to check in compartments with known outlets.
I doubt your battery is dead, as the AC still runs and the thermostat runs on 12v DC. Which means blown fuse for bathroom lights.
RV Cord ends/prongs can go bad, in which case you have to either have a new cord installed of cut off the last foot and install a new plug.
When running an AC unit off of 30 amp you need to make sure everything else is off. You never know when your converter is going to kick on to charge your batteries, your water heater is not on electric, your other AC of set to off, not just the temp dialed up. It could try to come on if the temp is reached, which would result in that breaker being tripped and possibly a hot cord.
If your RV is 50 amp adapted down to 30 amp plug you are not going to trip the main breaker in the RV. Which means you are relying on the breaker in your house for your man cave (30 amp? hopefully) it didn't trip and your cord got hot, which also means the wires in your wall going the 30 amp plug in the man cave could have gotten hot also.
