There's more than just a bad transfer switch, though that may well be the root cause. It sounds as though your chassis battery may be discharged or dead as well. (A transfer switch in and of itself would not cause any engine starting problems, since that's all 12V systems.)
It may not be the transfer switch itself that's bad, but merely a loose connection to it or between it and the electrical panel. With the power off, check that all the screws and wire nuts and whatnot are tight and that the wires don't look overheated. Check both the hot and neutral connections. (It would be a good idea to check the safety ground connections, too, but they wouldn't cause a loss of power if loose or disconnected, "just" a loss of safety.)