Unlike your thread title, none of the tires you mentioned are actually slicks. You did comment on the fact that the fastest ones have file tread, which work great on hardpack and dirt roads which don't have mud or sand.
So they will all do much better than a road tire. I ride my CX bike in the conditions you describe and I'm currently using a 700x32 Vittoria XG Pro. Next go around, I'll probably go wider up into a file tread tire in the 40mm range.
There's a big range of cost in the tires you talked about. The CX comp is an old design, heavy, and doesn't roll as well as other options but it would work and it's anywhere from 1/2 to 1/4 the cost of alternatives. You will be limited in tire selection since a lot of folks will be using 29er or 700c cyclocross tires for this type of riding. A narrow 26" tire is a pretty small diameter.