There are two thin wires from the trim switch to the solenoid one up and one down. Disconnect them from the solenoid on the trim motor and see what happens. Should one of those have juice all the time it means a screw or something has gone through the wire loom and caused a bad short.
If that's not it, go to the hardware store and buy a $10 multimeter. Leave the control wires off and the battery disconnected and use it to check if there is continuity through the solenoid. Measure from the big red thick cable coming into the solenoid to the thick wire going out of the solenoid to the trim motor input. Make sure the battery is disconnected when you do this. Solenoid might be stuck or broken and sometimes tapping on it frees it up.