It sounds as if your screen has failed - no light, but it responds (according to the sound you hear) when you touch it, so the touch-screen and the central CPU circuitry responses are still working, it's not that it is failing to operate at all, just no light on the screen.
It's probable that your Garmin Navigator is out of warranty (it is, if it is over 1 year from purchase date.) So it's a question, whether Garmin will service-repair this sort of defect in this sort of device (that may be some years old and no longer supported.)
Looking it up, on this page: http://www8.garmin.com/support/outofwarranty.html
it turns out that Garmin does still service-repair this device, for a flat-fee rate of $110.00 (that's what the page says when DEZL 560LMT is selected.)
If that seems rather a steep repair cost, be advised that other servicers are likely to charge as much, if not more - screen repair isn't cheap, and sometimes is not easy either.
Let me know what you think. I will advise you further.