Well, I found a solution that works for me.
I downloaded the Dolphin web browser, which is free. It allows you to download/save YouTube videos in an mp4 file that actually plays on the Droid (I've had a hard time encoding my own for the Droid, they never seem to work). It downloads fast and then I can save the video if I want or delete after watching.
I didn't realize I could do this in Dolphin, that is awesome.