Speaking very generically You have an up switch and an up solenoid and a down switch and a down solenoid.
Something has either energized the up circuit or the up solenoid has somehow activated or shorted.
There is also the possibility some part of the trim wiring is damaged or shorted.
Without the battery hooked up and the trim control connected
be sure everything is off. Connect a battery charger to the battery cables there should be an indication
the up is on if its a problem in the up trim circuit. Disconnect the control at the remote and try again.If nothing you know it is in the up button circuit.
I suggest the charger because connecting a charged battery into a possible energized circuit may cause a spark and explosion
of hydrogen gas at the freshly charged battery.
I suspect if the up solenoid discharged the battery it may now be shot and stuck on
whether the button caused the problem in the first place or not.