Lights are DC (12v) and outlets are AC (120v)
The surging could be a bad converter (powered by 120v supplies 12v.)
The LP alarm beeps when it has low DC voltage. The battery (12v) should take over when the converter is off. So it seems the converter cycles off but there is no battery 12v to take over.
If the converter is bad then it could be your battery has died and you didn't notice because of being on shore power getting 12v from converter. Some battery issues can make the converter act up.
If the outlets quit working at the same time the converter quits working, then you have a 120v problem as well.