I don't think you can do much about the handling problem with the tire you've got. You have three things working against your "on rails" quest:
outside temps are dropping
your touring tires are too hard to take advantage of outside temps (never reaching their operating temperature)
their normal/high tire pressures are putting less contact patch to the road
I have a similar problem with my BT-021 front. I put it on for longevity and it works well when it's 85+* outside, but in October temps it pushes in the corners like it's made out of plastic.
If you're serious, talk to your local m/c tire store about installing a softer compound tire for winter riding. Keep your 2CTs for summer.