A Cross bike will have a higher bottom bracket for a bit more pedal clearance if you do need or want to quickly hit the rough shoulder. Run some dual purpose tires and you would be set for pavement or the rough.

I'm not sure where you are or how mechanically inclined you are, but you might be able to find a nice used bike. I spent a week in Tulsa a few years ago and there seemed like a decent number of cyclists there so something used should be able to be found since the summer season is ending.