I understand that veterans are not required to take the freshman and sophomore level ROTC classes. Will I have to take these courses if I get a 3- or 4- year scholarship?

Yes, you will have to take these courses. As a scholarship winner you are a contracted cadet and must enroll in and successfully complete the basic course IAW the contract. As a 2-Year Scholarship winner or non-scholarship cadet your prior experience satisfies the basic course requirements.