I am not sure how nice of a mountain bike your Diamondback Overdrive is, so it is hard to say how a $600 hybrid (or any bike) is going to stack up against it for your use.
However, generally speaking, for urban riding, I would much prefer a bike like the Escape over the Overdrive.
Older hardtail and rigid mtbs do make great urban bikes, But I think the geometry of newer mountain bikes makes them less fun and nimble on the road. They work fine if it is what you have, and slicks do help, but something like the Escape is better, given the choice (IMO).
Edit: OK, I see that your current bike was only $500 to start with. Not sure how I missed that before. I think I have an idea of what it is probably like. I would definitely prefer the Escape.