I live in the same area and ride the same area (roads around the Hartshorne area).
I do my road rides on a cross bike with relatively nobby 35mm tires. since I don't come from the roadie side of things I don't know the difference between my cross bike and a "real" road bike. all I know is that compared to my mtb it is A LOT faster on the roads
depends what kind of riding you do. if youre gonna be on pavement all the time seems like a real road bike would be in order (if all pavement why bother with a cross bike?), but if youre gonna mix in some gravel (and particularly some off-road), and if youre not too concerned about absolutely maxing your speed on the roads, then a cross or gravel bike could work.
I like the idea mentioned above about a set of road wheels. I am think of getting a pair. most of my riding is mixed surface but I would like to do some longer road-only rides and want to have real road tires for that. switching tires is not ideal since I am tubeless (switching the wheels entirely would be easier)