i am a life long NYer, so I know a bit about living in the cold.
Yes, depending on where you live, there are some really cold days. Salt and sand makes your car a mess, but there are indoor car washes that take care of that. Schools do have fans, ac, and turn them on during the warm parts of the semesters. As for being constantly sick, the cold weather doesnt make you sick ( thats an old wives tale), its the germs and unsanitary practices. ( not washing your hands, not eating right, getting enough rest, too much stress etc) that helps get you ill. Oh, yeah, being around people who are sick, but dont cover their faces when they sneeze, or cough will do the trick.
If you are from the south, yes, the climate change will be drastic from what you are used to. Thats why I would strongly consider visiting the school in january or february so you will have a decent idea of what to expect, weather wise.( you are in the beginning of the spring semester, and there is plenty of time to make a final decision on where to put your seat deposit down). When you want to go out, just wear a coat, and check it ( or leave it in your car). If you are still worried about the cold, have a drink before you go. In the am, turn your car on ( make sure to leave the doors unlocked), turn the defrost on to high, and crank the heat up. That will warm up your car. There is no reason to winterize your vehicle, other than to make sure you have good tread on your tires, and to have a shovel and extra set of warm clothes and a blanket ( just in case).
Sure it wont be as pretty as Hawaii, or southern Cali/ florida in the winter. Its not as bad as living in the artic. Before you make your final decision, just go travel and see for yourself in january- that way you can make an informed decision.