Definitely Careful.....That is only a 15 amp transfer switch. You said you have a 50 amp plug. Things are not adding up. With your limited knowledge of AC power you need to be very careful.
Just looking at gauge of wire prior to the transfer switch its definitely not where the power to both AC'S etc passes through.
Once power hits the rear through the shore plug where does it go from there?
I'll check the generator and the extension cord, if they are both OK where should I check on the RV side of the shore plug?
It's a Forest River Cardinal 3825fl if that makes adifference