I think you have DOT and DR's backwards. I'm not the expert, but DR's are generally the "next step up" from a regular street tire. They just have a softer compound, and will get stickier than a street tire if you heat them up. Their tread pattern is not good for wet conditions and they are also bad in cold weather.

DOT tires are very much like slicks, yet they will have a couple of grooves in them that are considered "tread".

Like I say, I'm not the expert....but I think that's how it goes.