Get a digital multimeter if you don't have one and start testing the circuit, I think you have the right idea, just start checking any connection in the circuit that you can get to. Check for voltage where there shouldn't be any. Switches and relays, the actuator itself. Relays are cheap, I would replace that and go from there. As a last resort, you can always just put an on/off switch in the curcuit, it is not the best solution, but it will keep the battery from dying and would be easier than pulling the fuse.