I've been a verizon customer for quite some time. Had a droid1, and never really had any issues. I can't say that it never locked up in the 2 years I had it, but it certainly wasn't often.
Now I have a Droid3 (arguably the most awesome phone on the planet!)
I've had it for a couple of months now, and it hasn't locked up on me yet.
My husband (still with his Droid1) will occasionally have more problems, but I am personally convinced that it is because he will NEVER reboot the thing. When he tells me he is having a problem, I ask him... when was the last time you shut it off/turned it back on??? usually the answer is over a week...
I change batteries usually once a day or every other day, and swap them out since I have an external battery charger, so my phone is shut down/restarted every day or at the very least every other day.
I have a task-killer app, but I don't use it, and don't have it set to "auto-kill" anything.
The Droid2 was supposed to be a really awesome phone... and for Verizon? If you get a bad tech support rep; hang up (preferably while you are in the middle of a sentence so as not to appear rude...

) and call back for another...
Verizon is expensive, very "big brother", however they have been very competent, their coverage rocks, and they have been reasonable with me a couple of times when I've made somewhat unreasonable requests... I compare this to my experiences with AT&T, Cingular... ("Please Kill Me Now" is what comes to mind when I think of them...)
All that said; you may simply have a defective unit, or need to restart it once a day.
If that doesn't work, see if they'll trade it out, and if not; if you have insurance... God forbid that something would happen to your Droid2, but if it ever did, you could perhaps get a relatively cheap replacement Droid2.
Or... perhaps just upgrade to a Droid3. I can't say enough about how much I love this phone...
Good Luck!