If the transfer switch is not actuated, the circuit between the pedestal and circuit breaker panel is open, so you would read no continuity. Jim A commented that if there's no generator, the transfer switch would be locked down for shore power - presumably with a mechanical device so it doesn't need to actuate the contactor. But when I replaced my transfer switch (several times), none of them came with any kind of lockdown device, so I'm not sure what you should expect there.
Next step is to take down the basement wall to get to the transfer switch. Location may vary a bit by floorplan, but it's probably behind the doorside section.
I CAN'T EMPHASIZE ENOUGH THE NEED FOR EXTRAORDINARY CARE WHEN WORKING WITH THE TRANSFER SWITCH. MAKING A MISTAKE COULD BE FATAL.
On one end of the transfer switch, there's a single cable. That end goes to the circuit breaker panel. Directly across at the other end of the box is the shore power input cable. Next to the shore power input is the generator input cable.
With the power cord unplugged from the pedestal, you can safely open the transfer switch enclosure (one hex screw). If you manually depress the contractor assembly on the shore power side, (UNPLUGGED FROM THE PEDESTAL) you should read continuity across the switch.
If you are very careful, you can leave the cover off the enclosure and plug into shore power. There's a wiring diagram on the inside of the cover that shows L1, Neutral, and L2. With power applied, measuring the input side should read 120V between L1 and Netural, and also between L2 and neutral. You might also check between the hot wires and ground to make sure you don't have a bad neutral. Normally if you read 120V between L1 and neutral, you would also read 120V between L1 and ground. If there's any significant difference, you might need to investigate further.
If the voltage is correct on the input side, and the transfer switch is working, you should read the same on the output side, and at the circuit breaker panel.
If voltage is low on L1 for any reason, say below 105V, the contactor may not actuate and the coach will not get power.
If the contactor doesn't actuate, or has contacts that aren't conducting power properly, that would explain your problem and you'll likely need a new transfer switch. About $150.
It's a pain to install a new switch because you'll be lying on your side with feet hanging out the hatch. It's hard to reach the parts and the #6 wire is difficult to work with. Paying $100-200 for someone else to do the work might be worthwhile.
If you do it yourself, before powering up the coach with the new switch, turn off all circuit breakers except for one that feeds only outlets. Don't have anything plugged in to the outlets. Test the readings at the outlet to make sure you didn't get wires mixed up. If you made a mistake, this should keep you from frying all your equipment.