Yes, a hand held meter is a necessity, IMO. If you don't know how to use one, the learning curve is short and will save you many hours of frustration and $$$.
If it was my RV, I'd disconnect the various failed appliances and plug them back into an extension cord, connected into a known, good, 120V receptacle. That is not one in the RV with the generator supplying power and not a campground pedestal. Find a business or homeowner connected to grid, measure the receptacle's output, and then connect the extension cord to the appliance.
A microwave will probably have a fuse inside that may save the microwave. A TV does not have a fuse like that.
When you get home, I would hire an experienced tech to look over all of the devices connected to 120 V and also the load center and shore connection bits.