I agree with Jcleppe on "a bird in the hand", but much has to do with the quality of the bird. If it's a turkey but you had your heart set on goose, then perhaps you should skip the turkey. Nevertheless you should think hard about declining a scholarship unless you and your folks are close relations of Warren Buffet. When you're hungry a guaranteed meal of turkey is more filling than a hypothetical goose.
This is just my opinion really as I don't have a whole lot of insight on this. However, my impression is that as long as there is room in the unit, you will be accepted. Since you won the AROTC scholarship I'm confident you meet the minimum requirements. There is an application that must be completed which I'm sure you can find online (at least in general form - see here to get an idea:
http://www.sc.edu/nrotc/content/Documents/collegeproapp.pdf). You will also need to complete a physical with your own doctor that is similar to what you might do to participate in a high school sports team.
If you plan to pursue this then you should contact the NROTC unit at the college you will be attending ASAP. They can give you more info and probably a better heads up on the chances of being accepted. You can also get the ball rolling on all this including finding out which classes you will need to enroll in at your college, which you will no doubt do early summer as part of your freshman orientation.
Keep in mind there are NO guarantees. Competition for in-school scholarships are national and the
average stats of awardees are equal to or better than that of USNA appointees. If you don't win a scholarship then you must achieve advanced standing by your rising junior year. Again, that is a national competition as well. Much will depend upon the needs of the Navy at each step. You will have to work hard but probably no harder than if you wanted to excel in AROTC.
I'm sure you've discussed this with your parents, but if you haven't, you should. Finances are important and can trump all, but I do understand about following your heart.
Do you plan on pursuing Navy Option or Marine Option? I primarily ask because Navy Option will require 2 semesters of Calculus and 2 of Calculus based Physics. These subjects aren't everyone's strengths so I point it out in case it influences you decision. The only math and science requirements for Marine Option are those imposed by your college's core requirements.
Good luck!
