The same problem happened to my D1 last week!! Your digitizer has broken. The only solution is to replace it or buy a new phone (If you do a search on "Droid" and "digitizer" you will find many people who have had this same problem)
I researched repair stores and finally settled on a place called Mission Repair in KS (
www.mission repair.com). Sent my precious Droid to them on Tuesday and got it back as good as new on Friday for a cost (including Fedex) of around $90 - certainly a lot cheaper than a new phone.