The tilt sensor is just for the tilt gauge in the front of the boat, it wouldn't keep the system from working. And because the solenoids are clicking, that means everything foward of the solenoids is working, and the problem is either the solenoids or the trim motor.
This is actually fairly simple to troubleshoot. Mercury uses 2 different style trim motors, a 3 wire motor and a 2 motor. The only difference is the 3 wire has a ground on the windings, but the rest of it would be the same. So to test it, what you would do is disconnect the large blue and green wires from the trim solenoids. These are the wires that run down to the actual trim motor. If you do have that 3rd black wire, just make sure it stays grounded for the test. All you do next is touch blue to battery positive, and green to negative, and the motor should trim up. If you switch them, and put blue to negative and green to positive, it should trim down.
If it works replace both solenoids. (as you would replace them as a matched set)
If it does not work, replace the trim motor. As the trim motor has either burnt out, or more likely, the oring seals failed and it got water in it and it rusted out.
Does that make sense? If it does, then that should do it. And just let me know if the ratings box gives you any problems or if you can not see it. If it does not, or you need more detail on this, by all means feel free to ask before doing the ratings. But thats all you have to do is jump the blue and green to the battery and see what happens.
If I do not hear back from you, thanks in advance, and good luck!