You should hear a "clunk" inside the switch after plugging into power as the relays switch to shore power. If not, something's definitely awry. Here's a link to a good write-up on your transfer switch and how to repair some issues:
ATS5070 Generator Automatic Transfer Switch
There are less expensive options as well.
Mine went bad in my 2002 Suncruiser and I decided to just replace it since it was pretty old anyway. I used this one:
Just make sure you label all your wires before disconnecting and take photos as well.
Here are two things to try first:
1. With shore power unplugged and generator off, carefully loosen and re-tighten each connection in the transfer switch. Sometimes this will get things working again.
2. Make sure your shore power cord is OK and that power is actually getting to your transfer switch. You can do this with a multimeter while plugged into shore power by carefully measuring the voltage between the black and white shore power wires where they connect to your transfer switch. You should see something between 110V to 120V or so. Make sure your multimeter is set to read AC voltage.