On your particular model, your transfer switch is attached to the backside of your main circuit breaker panel. It is very difficult to access and won't reveal much unless you want to take the panel and switch completely out of the cabinet. There's nothing that you can check without removing the panel and removing the cover on the transfer relay (switch).
It sounds like the contactor (relay) points were temporarily welded together. This sometimes happens when the relay is switched under a heavy load. You should always remove the loads before switching (turning the generator ON or OFF) or plugging into shore power.