TT - I don't think your problem is in fuses. The shore power cord being plugged in should not cause the top to raise or lower, because it doesn't supply enough power. The top should ONLY raise on battery power. If you disconnected the battery, I doubt having the shore power cord plugged in will allow you to raise the top. I suggest you try that and see if I'm being truthful.
One more thing - do you hear anything when you move the lift switch to "UP" with only the battery connected? The lift switch actually energizes the solenoid on the lift motor, and THAT solenoid switch, when it closes, completes the heavy wire circuit to the battery that allows full battery power to flow to the lift motor. Do you hear a "click" in the motor area when you move the lift switch to UP? If so, I think the wiring from the battery to the lift motor itself has a poor connection, possibly due to corrosion, or, something may just be loose. If this is the case, having the shore power cord plugged in may just supply sufficient power to help overcome the faulty battery connection. It DOES add to the battery's voltage, which can bridge a weak connection.
- Jack