How old is your rear tire Dave? I just replaced mine and the difference was dramatic ! When the tire gets worn even a little, the center becomes flat. so it's like a car tire with square edges. Of course the degree to which that happens is gradual so you don't notice it happening. When i recently put on a new back tire the bike felt like a sports bike. Now i'm taking turns like a sportbike leaned way over and it falls right into the turn with no effort at all. It responds to my thoughts, and it grips like it's glued to the road.
Now why one side seem to go before the other i have no idea unless each person has thier own particular routes they ride a lot which have more turns to one side or the other. But you said it's BOTH sides, so i would suggest looking at a new tire if yours isn't close to new. Mine feels like a whole new bike. If i could afford it i'd put a new tire on 1/2 way thru the wear for now on ! By the way, i think this affects us more than a lot of other bikes because of the tires size/shape. Sport bikes for example have those low profile tires that are flatter from edge to edge than ours, so even at high wear they wouldn't develop a squarish shoulder.