Are you a Freshman? Many schools do not allow Freshmen to have cars on campus. Even if they do, can you get parking? Is there a waiting list? How much is a permit?
And you will need car insurance for a car you will not be driving much, so that cost adds up.
All that aside, on to the drive.
Obviously you are way behind schedule here and most Freshmen report to campus at least 5 days before classes start. Not being negative, just stating facts.
2700 miles. If you can drive 16 hours a day, which is about 900 miles, you are looking at three days of driving. That is a very hard schedule even for a professional driver. So if you do 12 hours a day that is a 4-5 day trip.
I have driven 8 hour trips all the time. And some 12 hour trips, not easy.
You have about $250 in gas, food, and you have to get a hotel room at least 2 nights. You can nap in the car, but for drive you will need to get some sleep.
You would need to get the car FULLY serviced and checked, oil change, coolant flush if over 3 years since last flush, check all fluids, belts and hoses. Check and inflate spare tire.
Then you have breakdown risk. No way to know, but you are driving a 10 year old car, could a water pump, alternator or some other wear item go bad? Sure. No way to say is that a 1% chance or25% chance. That would require extensive miles of towing and would add 2-4 days to your trip.
My opinion - buy a plane ticket, pack two suit cases, pay the baggage fees, get some 14x18x12 boxes pack some things you need, then have them shipped to you once you get to GT.
Best of luck.