The Liberty transfer switch if working properly prioritizes power source.

If it does not sense shore power or the generator output it defaults to inverters, and only the 120V circuits powered by the inverters such as TV, Refrig, forward and rear outlets, microwave, and water pump will have pwer to them.

When connected to shore power the transfer switch makes that power available to all circuits in the 120V panel and the inverters are fed 120V power and instead of providing power to the panel they function as chargers and they pass the shore power through their own internal transfer switch to the previously mentioned circuits.

If the transfer switch detects the generator running (output from the generator) it disconnects from shore power and instead transfers generator power.

With everything working correctly in the transfer switch no two power sources can ever provide power simultaneously, nor can they back feed. If the prongs on the power cord were burned my first guess will be there was corrosion on the prongs or in the shore power receptacle which created resistance and in turn created the heat to burn them. It had nothing to do with the generator being run, but more likely was due to a large current draw such as 40 amps or more prior to or after the generator was run.

As to the charging issue with the house batteries, my first place to look for why charging did not occur would be with the inverters. I doubt they both failed simultaneously or suddenly started working again, so I would look at switch positions and verify that 120 Volts was reaching the AC panel. More information is needed. BTW, the appearance of charging voltage is not necessarily time related unless the inverters are set up to go into equalize mode every time they are powered up. That is an 8 hour time and should only be done when required. All other charging is at the 14.2 to 14.4 rate for bulk and absorbtion, and then a lower voltage for float. The times within those three phases is based solely on the battery state of charge.