Looks like could be several issues.
1st. Check the 120VAC voltage at the pedestal. If wind was bad, power to campground could be out. If power at pedestal, then burned plugs could mean loose connections causing heat. Have campground check connections in pedestal breaker box. You can get replacement plugs for your power cable. Depending on what is burned/melted, you may need to replace both plugs and sockets.
2nd. If you lost all lights once the power went out, you most likely need to replace your battery. When connected to shore power, the converter was powering your 12V lights. Once the power went out, you battery should have taken over, but if you had no lights, the battery was bad and needs to be replaced.
Did you have high current items running, ie electric heaters or coffee makers. It is possible that you were drawing high current through a high resistance ( poor) connection causing heat.
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