Which type of bike for me?

[KnottedYet]

with your budget you could buy two Surly LHT, one for nice weather, and one for rainy/snowy/yucky weather.

just sayin'....

More bikes, more better!

(oh, and you get them at your bike shop already assembled. Usually if you want a bike your shop doesn't already carry, you order the bike through the LBS and they assemble it and sell it to you at MSRP. They get the profit from the normal mark-up so there is no additional expense to you. Surly's website lists dealers who already carry their bikes, and that might be the way to go if there's one near you. Surly isn't as common as other brands, but they have a bit of a cult following. Terry bikes have the same thing going on, with a list of dealers who can get you the bike, or you can have your shop order one.)

As far as riding through the park: the LHT would do fine. Very fine. My LBS says the frames of the Cross Check and LHT are quite similar and withstand similar forces (cyclocross and loaded touring). They are sturdy dependable beasties and won't faint at the sight of a dirt trail. I've ridden my CC on all kinds of stuff (including no trail at all!) with no complaint.

Any of the bikes designed for loaded touring (which are more likely to fit you in standard sizing than a cyclocross bike) are going to be strong frames with strong wheels and likely have rims to take wider tires (like the 28mm and 32mm I have on a couple of my bikes). All together these make the bike able to handle a variety of surfaces and situations. Fine on trails, grass, dirt, asphalt. The 42 cm LHT comes with 26 inch "mountain bike" wheels, anyway. (the ones I've seen had road slicks on them, but you can ask your shop to put juicy wide MTB knobbies on if you want)

Sturdy All-Rounder Tour-Beasties:
http://www.trekbikes.com/us/en/bikes/2008/road/520/520/ Not sure what size you'd be
http://www.jamisbikes.com/usa/bikes/...08aurora.html# Very popular, but I think they don't have small enough sizing
http://www.konaworld.com/08_sutra_w.htm Another popular one, but again a sizing problem.
www.bianchiusa.com Could you ride a 44cm? The Bianchi Volpe comes in 44 cm.

These will at least give you some ideas. Kona made a "Jake 2-4" for a while, a small cyclocross bike with 24 inch wheels. Maybe check eBay or Craig's for one.