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Your Biggest Advantage
As a former Maine-iac, can tell you that the others here are right. A good, relatively new pair of all weather tires is all you need. 'Snow' tires don't offer that much more tread and studded snow tires have been outlawed in many states because of the damage they do to the roads. Your ace-in-the-hole though is the fact that Corollas are front wheel drive which is much better on snow and ice. Just make sure that when you pick up you car that the all weather tires are on the front. I've had a front wheel car of mine (a SAAB 93) returned to me with them on the back! But then front wheel drive cars were a rarity then. The other very important bit of advice is to slow down in snowy weather. When you think you're going slow enough, go slower still. And if the roads start to ice up, call it a day and head for a motel.
I also just got back from Washington DC and the beltway there, I-495, was nothing more than a huge circular parking lot. I've had similar experiences in Philadelphia and of course New York City. I'd try to thread the needle between the large eastern cities by taking I-65 north out of Nashville to Elizabethtown KY, then the Bluegrass Parkway east to US-60 around Lexington KY and I-64 east to Charleston WV. Then it would be I-79 north to Morgantown and I-68 east to Hagerstown. Finish up using I-81 north to Binghamton NY, I-88 east to Albany and I-87 north to Lake George.
Although the above sounds complicated, it's really not. Draw it out on a map and you'll see. While it is about a hundred or so miles longer than the shortest possible route that mapping routines will give you, it will be much more relaxing, scenic and enjoyable.
AZBuck
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AZBuck's suggested route makes good sense. I just completed an I-81 segment from Maryland to North Carolina, and it was terrible. Tons of truck traffic, more than in past years, which just made it a high speed stop and go experience. Not much scenery to enjoy in those conditions. A truck was continuously viewed pulling into the fast lane to pass another truck that would often require a good couple of miles to accomplish. We had to detour at one point due to a total road closure and drove a couple of US-numbered highways which was much more relaxing. I-81 north of Maryland to Binghamton, NY, should be fine, but I would also consider driving I-79 from I-40 north to I-80 across NY.
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You could consider driving most of the way across on the summer tires and then getting new tires at a chain store somewhere along the way. The shops are pretty quick about it - especially if you make an appointment ahead of time.
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One thing not mentioned - is there a particular reason for having Lake George as a target point?
The reasoning behind that being that much of the Lake George proper area is things that are not open in the winter, including even many of the motel type lodgings. Even McDonalds (though it is now permanently closed) used to close for the winter when it was around - not something you see too often these days (unlike in their early years).
It can certainly be a base area if you want to explore the more natural things (including many options for skiing of both downhill and cross country), and there are other places that are reasonably close by (reasonably being within an hour or 90 minutes in different directions).
Just wanted to be sure you didn't get all the way here only to find not much available, given that some areas (such as where you are in Las Vegas) are much more year round and might not be something everyone would think of about other locations :)
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If you buy new all-season tires, put them on all 4 wheels. I'm surprised the dealer sold you tires that are not all-season, the vast majority of tires these days are all-season except high performance tires for "sports" cars.