i go on relatively level hard pack dirt/clay all the time with pure road tires in x32 or so. you really only need tread for climbing steep hills on loose surfaces. And, in deep sand, soft mud, etc, the only thing that helps is a really big/wide tire with open knobs.
the cyclocross tires I've seen have had rather soft rubber, which will wear out pretty quickly on mostly pavement riding, and are best reserved for actual cyclocross racing.
I'm not familiar with the bike you have, but I'd go with the biggest tires you can mount comfortably on it, like if it will take x35 or x38, go for it. even pure slicks in the fatter sizes, running lower air pressure, work great on gravel etc.
I've used Specialized tires for years, but have been rather disappointed in their tires these past few years, and am not likely to buy any more of them. I'm really happy with the Vittoria Hyper Randonneur's I put on my hybrid, in 700x32. Also really like the Continental Gatorbacks I've been putting on my son's road bike, and the various Panaracers are a great lower budget tire (Pasela, etc). I keep hearing that Schwalbe makes great tires too but I haven't personally tried them.