Good to hear that the 244's wear a long time on the rear, will have to be careful about mud though it sounds. Are you running the 244 on the front? Why is it such a hateful tire?
When I first researched Shinko, I opted for the 700 series partly due to research and partly due to the fact that I didn't trust a tire that was so cheap as the 244. As I stated the rear didn't seem to have the traction on dirt and gravel, but the front was miraculous for gravel road stability.
I had just gotten used to the mild headshake on gravel over the years and figured it was just the way it was, not harmful but irritating. I tried the Kinda 270's at the recommend of a friend who claimed to have that problem on a KLR which was cured by that tire.
In my case the 700 is the cure for that on my machine. The front feels much more stable and planted with it on my gravel road wanderings. ( I haven't had the opportunity to test it in the mud yet) It seems good on the street too, not upsetting my confidence in the twisties.