Start off by replacing the up solenoid on the motor. I will try to take a pic tomorrow of mine so you see what it looks like. Hard to describe but their are 2 solenoids(up&down) square in shape, mine were on the right side of the motor.they have a wire harness that plugs into the bottom of them and they are secured to the motor by a small bolt. Sorry not mechanically inclined to give a better description but I had the same problem on my 2000 40hp mercury and after talking to a marine mechanic and him showing me where they were, I replaced both of mine. If that doesn't fix it then i would suggest replacing the wire harness. That was the issue with mine. $50 for the harness plus labor and it was fixed. Hope this helps and I will try to get a pic of the solenoids tomorrow.
Tom