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.