I had the same problem on a Nuvi 1490T in the middle of a New England vacation. After checking with Garmin Customer Service, the unit that wasn't even rebooting had to be replaced by a refurbished unit.
You should check with Garmin CS. It takes a while but the service was good.
The Garmin Nuvi 3760 was produced with the plug in the back of the unit.
When you do any updates from your computer or simply (as I have done) unplugged it to store it out of sight in your car.
Each time you unplug and replug the device you are compromising the plug which causes it to shot and be problematic with the screen.
You may need to take it back to Garmin for repairs.