As far as I know, the Finest is still a true road bike, even if it does have disc brakes now. (Discs are becoming more and more common on road bikes.) I really like the look of the Giant Invite 2 (if Giant's geometry actually agreed with me, I'd get one). I also like the Tricross (it's Specialized's version of my Charge.) No experience with the Trek, so I can't say anything about that. Each company does things a bit differently, so if you can possibly try it out, do so! Most bikes made from metal (so, steel or aluminum) have the ability to take a rear rack, although fenders might be somewhat more difficult. I'd take the trip, but call around to see if they have them in your size, since finding bikes, especially cross bikes, in those smaller sizes can be difficult. In a road bike, you'd probably ride a 48 or 50cm, but cross bikes run larger because the bottom bracket is higher so you don't clip things. Again, call around.

Some advice for your current bike: If you ride a lot, you will chew up your bottom bracket, especially if you get water into it. They're not terribly expensive, so even if you plan to sell that bike at some point, replace it. Replacing parts for comfort and performance is pretty much part and parcel of bike ownership. I've replaced the saddle on all three bikes, the roadie multiple times. Unless some miracle occurs, you'll have to do that with your new bike too. For the flat issue, make sure there's nothing stuck through your tire like glass or a sharp piece of gravel (the bits that come off painted lines are actually really sharp!), and then double check that there isn't a sharp spoke end that could puncture a tube.

Just a note: I know people who have taken cyclocross bikes on non-technical (or even technical) mountain bike trails. They are both crazy. I've done it too, once (my post is on here somewhere), and came home with a load of bruises. A CX bike is well-suited to dirt and gravel roads. Not so much mountain bike trails--the lack of suspension and the geometry of the bike make a big difference. It's doable, but won't be less-frustrating than with your FX.