Hi--I'm "Dewey" and I just upgraded to a Motorola Droid from Verizon as of last Saturday(6/5), and I think it was the new Droid with OS 2.0 (not exactly sure since I'm at work and my phone is not with me). I joined this group to get a better understanding of my phone,and to troubleshoot. I had a Blackberry Curve 8330, and really liked it, but went to the Droid because of all of the free apps.
I am having the "screen goes black problem" as well, and I have a hard cover (body glove brand to be exact). My Droid is tempermental in that sometimes I can get the screen to come back up by pressing the power button/sleep mode, but sometimes I have to slide the keyboard too.
Somwhere on this site, I saw where to install either Screebl or another app (which the name escapes me right now) to keep the screen from doing this. I also got rid of task killer too. It still does this though.