It may be just me, but I really hate carrying things on my back. I mean, a Camelbak is okay, but anything more than that really bothers me. I very much doubt you'll want to carry a tent on your back. While you can find very light ones, they also tend to be long, and it will interfere with your helmet and you'll find you can't move your head without hitting either the tent poles or the tent itself. Leaning on your hands for five or six hours can be troublesome enough, you don't want to add another 20 or more pounds on top of it. And food, unless it's that dehydrated stuff, can be very heavy.

But even if your bike doesn't have eyelets and things, there are always bolt-on clamps you can use instead. These often don't hold as much weight though. EG, most seatpost mounted racks max out at about 25#. The ones that bolt right to the bike don't really have a limit.

spacejack is also correct saying the X-C bikes have a higher BB, but unless you're racing, that won't have much effect. A high bottom bracket lets you ride over larger rocks without bottoming out, and it doesn't sound like that's the sort of stuff you'll be doing. X-C bikes will sometimes also have suspension forks if you like that sort of thing. It may also have a disk brake option. But that would be over your budget and not really necessary for touring. V-brakes are nice, though, especially when you're loaded down.

Really, there are a boatload of choices. A touring bike might also work for you if can get one that can handle 700x35 tires with fenders. Or at least 700x28. I'm not sure I'd want to go off road with anything narrower than that.

Cyclocrossworld is a good resource, as is Road Bike Review and the companion Mountain Bike Review. Lots of candid opinions there.

Provided they're a good one, your LBS might have the best advice.
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Science says, "YES!"
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