A drama script is not necessarily something that is dramatic. It just means that this piece of literature was written to be performed on stage or in front of an audience. So play scripts certainly fall under this heading but so do movie scripts, radio dramas, and even stand-up comedy routines if we can find the right sources. That does not, however, mean that I would like to see us fielding a flock of George Carlin impersonators. For the most part, DI scripts are play scripts.
The decision as to whether to use a monologue or a dialogue script is completely up to you. Both have seen great success, both have won nationals, and both are out there to find. My first concern is that you find something you like. DI often sounds a lot like prose to me. We generally allow for a little more movement and greater acceptance of pantomiming gestures. DI is an event where successful performers really concentart on character development.
It is not my favorite event. This is probably due to a couple of bad experiences I have had judging DI. My first year out of grad school I had to judge a DI round of seven competitors rather than the traditional six. It was the last round of the day and 4 of the 7 went over three minutes over time. To make matters worse, they all knew just one way to express an emotion and that was to scream. The room was large and plaster-walled so I heard every scream four or five times as it bounced off the walls in the room. I left the round physically shaking from auditory overload. A couple of years ago I had to stop a student who just passed the EIGHTEEN minute mark in her DI. And she looked at me as if I had just stepped on a fluffy kitten for doing it.
As with most events, finding the literature is the hardest part of doing drama.
I would also encourage you to read this basic oral interp at the college level article