Hi Doug, your generator may have an auto reset built in, and it maybe just burned out. If it is a newer 12 volt system generator, with reversed pole protection it will read 0 volts, if read open! These chargers will only apply voltage to the leads when a battery is present.
I own a coleman 1800 watt and the 12 volt system is really only good for bring your car battery up enough to start. The open voltage on my generator's 12 volt system is 13.6 volts and will not fully charge even a car battery.
You need 16.5 to 18 volts minimum open voltage for charging. (unless you are using the reversed pole protection kind, in which case you can't read open voltage). If the 10+ volts is open voltage on the sears charger, you need to buy a new one!
Blessings, Snow Crow