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Might be the transfer switch is bad..............unless you are using a 30 amp adapter on your shore cord. Then you lose about half of the coach power. Many people don't know that. You can test the transfer switch with a small digital multimeter. About $20 or so at most hardware stores and easy to use. I can direct you in that. All follow ups are included in our fee.
Check for power on both sides of the shore cord INTO the transfer switch with your meter once you get the cover off. The hot legs are black and red. You use the white leg for the other lead on your meter. IF you have good voltage IN then do the OUT side and see if the problem is there. This will tell you if the problem is the cord or the ATS. Some you can buy parts for and some not, depending on who made it.
It may be in the plug itself........but, check to be sure that the source isn't the issue. IF the cord doesn't go clear up to the ATS, then check that connection.
Does the shore cord go all the way to the ATS? Most do. And you are correct that it may fail under load even though it shows good continuity.
Rare that it would be in the cord.....more likely the plug.