Both of my previous gassers had 2 salesman switches, one for coach batteries and one for chassis and I would just use that for storing it during the winter and never had problems. I guess I never had any other draws when it was shut off like you do. You can do what Bob suggested but I would try to find the draw as well.
I actually have the opposite problem from you where my coach batteries go dead if I'm not plugged in and forget to shut off the inverter. My steps work off the coach batteries as well so maybe yours do too.