Yes, you should be able to buy a car and car insurance without having a social security number. While some insurance companies require an SSN there are others that do not.
Insurance companies will require you to have a US driver's license, which you state you already have so that should help you be able to obtain the auto insurance that South Carolina will require you to have on the car you buy.
On CarInsurance.com's website SSN is optional during the purchase process with some carriers and required with others. If you do not have a social security number and it is required then you can call our service center and they will assist you so you can purchase a policy. You can talk to an agent or our customer service team over the phone by calling us toll free in the United States at 1-800-964-4859.
To find out what documents a dealer will require you to have when purchasing a car; speak with a representative of a dealership. You can contact the South Carolina Department of Motor Vehicles for information on how to register and title a car in SC once you have purchased it and placed insurance on it.
If you need advice on how to buy a car in the United States you may see if your college or university's student affairs department can give you advice. Some places of higher education have guidebooks for international students which includes information on how to negotiate and buy a car.