I'm betting on the digitizer. I'm thinking about going to that one youbreakifix website for a fix, but a screen replacement is about $100 for Droid 1.
-EDIT-
It's $40 for a digitizer mail-in "Do It Yourself" from ubreakifix.com, $100 for a full glass/touch screen replacement (shipping not included)
What sucks? In order to get to the digitizer in your phone, you have to take EVERYTHING,
EVERYTHING apart from the phone. The Digitizer can only be reached by working your way from the back to the front.
[video=youtube;e3zQIxhGxdM]http://www.youtube.com/watch?v=e3zQIxhGxdM[/video]
You can watch the whole video, but here's the thing: Pay attention when it gets to 7:48. That's the piece you need to repair / replace, that's the digitizer.
I also sent ubreakifix.com an email, asking if they are able to replace just the digitizer instead of the entire touch screen. They haven't emailed me back though, I'll keep you guys updated, I know there's a lot of people with this same problem who probably, like me, have an expired warranty so you can't ask for a replacement phone unless you go pay for it.
And this won't be my last edit, I'll constantly come back to this post or add another of somebody else makes a reply. The link below is the digitizer from ubreakifix.com, DIY repair
http://www.ubreakifix.com/motorola-droid-1-glass-digitizer-404.html