Heaven knows I'm not a skier...but I took a look at my region for you. There are lots of places close to some of those resorts. Only a couple that are "onsite" though.
Saddleback--Rangeley Lakes Resort. It is more rustic than many places, but has individual cabins, which is kind of nifty. I'm staying there for the first time this summer. Google maps says 7 miles between Saddleback and the resort.
Black Mountain in Jackson, NH would be close-ish to both the Attitash resorts in Bartlett and anything in North Conway. (10 miles or so)
Waterville Valley--this would be my pick. There are 3 timeshares in Waterville Valley. Inns of Waterville Valley will likely be the easiest to get in. I stayed there last Feb break, and it was fine. Old, but truly fine. The ski resort runs a shuttle that comes down to get you...you wouldn't have to use a car. WV is like a whole little village. Loved it here!
Cannon in Franconia, NH has a reputation as being more for the expert. However, there is Mittersill resort pretty much at its base. I've never stayed there, and it's likely for someone who isn't looking for much out of the resort...more a place to hang one's hat. Lincoln wouldn't be far down Rt 93, though...and there are several timeshares there.
Bolton Valley in VT has a timeshare as well. Bolton Lodge, I think, or maybe The Lodge at Bolton Valley.
New England places tend to be in a league of their own. They are older, and more about the location than the "extras." No lazy rivers here.
They'd freeze up anyhow.
Enjoy picking a new place or two to discover!