I am looking for replacement tires for my 2012 STX 4X4.

Several people recommended Cooper Discoverer ATP or AT3. The ATP's are impossible to find in my area. If you can find the AT3, they are about $100 more per tire than at Discount Tire. (The closest Discount Tire to me is about 300 miles away.

A local shop near me is recommending Hankook DynaPro ATM. Are these tires any good?

 

I just put a set of the Hankooks on my 2012, still too new to tell whether they are "any good". All the reviews on the interwebs are positive though..

 

Discount tire will ship them to you. I just put in Cooper A/T3s and Harrisburg for the zip and it came up free shipping and $40 online order discount.

 

Call DiscountTire, I believe they can send the tires to your prefer local shop for mounting. Or you could consider Open Country AT or ATII, way more aggressive and wear good, check them:
http://4wheelonline.com/Toyo_Tires.202664
http://www.treaddepot.com/group/opat2.html