Actually the Ag tires do a lot less tearing up the ground because they don't slip,. They will leave some marking in soft soil but nothing I can't live with. The improvement in traction is amazing especially filled. Slipping turfs did much more damage to my place. You need more fluid as well. Normally put the valve on top and fill to the valve then air up. I used windshield washer fluid. 8 gallons in each rear for me. Best is probably Rim Guard. It's a brand name for beet juice. Doesn't freeze and weighs a lot more per gallon.
http://www.rimguard.biz/
I'm running Carlisle Tru-Powers in 26x12x12 and love them. Also have Vredestein v61 ribs on the front. Filled at all 4 corners.
This is a great place for tractor tires. Great customer service and carry tires others don't.
http://www.millertire.com/
Some folks are running Heavy Duty All Purpose (HDAP) tires with good results. More aggressive than turfs but less than AGs.
http://www.carlisletire.com/products/at ... atalog.pdf
I did some looking and you have kind of an odd size. Choices are mainly turf and AG lug. Neither comes cheap in that size. I think what you have is probably better than most turfs. Turfs are made for mowing on dry flat ground. Add hills or water and you find their limits very quickly. I'd call me miller above and ask for suggestions. They specialize in hard to find tire. Sort of the Coker tire for tractors.
One other thought is you might find a truck tire like a Mud Terrain to fiit that 16x9.50 rim that is the right diameter. Some folks have been known to do that. Run some Swampers or LTBs on your tractor.