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So the drive time and drive distance are fine. The issue is the straight line distance being calculated due to the tool configuration. With the way you have the distance tool set up - you're measuring two different things.
Start Point in Charlotte to the centroid of the disaster area via drive time/drive distance.
Start point in Charlotte to the nearest edge of the disaster area via straight line.
Notice that only the drive time/drive distance option mentions centroid based calculations.
It may seem a bit unintuitive, but the way the Distance Tool works when measuring straight line distance from a Point-to-Polygon (and Point-to-Line) is to use the nearest edge.
If you want to calculate the straight line distance and the drive time/drive distance to the centroid of the polygon - you'll need to use a spatial info tool to generate the centroid point first and then set that as your destination point.
Off the top of my head there are a few work arounds to try to accomplish this. You could build an iterative macro that generates successively larger drivetime trade areas from your starting point until it meets the condition of intersecting the disaster area polygon. You'd have to use the option in the trade area tool to take the drivetime value from a field - then use a formula along the lines of [Starting Drivetime]+(([Engine.IterationNumber]+1)*10) to increase your successive drivetimes by 10 minutes.
Not having the DriveTime datasets, I can't test these options, however, I want to ask a clarifying question:
Are you looking for the minimum drive distance, or the drive distance to the closest point via DistanceMiles?
If you are looking for the drive distance to the same point identified by DistanceMiles, you may be able to accomplish this with two sequential "Distance" Tools. The first would only output the Distance, and you would enable the "Create Nearest Interior Point inside of the polygon" option. Then, in the second Distance tool, you can match to the new "InteriorPt" field and enable the DriveTime option.
If you are looking for the minimum Drive distance, you could use @BenMoss' suggestion of the Poly-Split tool to create the outlining points, and then a Find Nearest tool to find the closest point, configured for Drivetime.