Does non-specific scholarships mean that the check will be written directly to the school, and does not have to be used specifically for tuition, books or room and board? If that is the case, I believe USNA will accept checks that will be deposited directly into the mids account to be used for uniforms, etc. There is a document on the USNA website that talks about a mids budget and also discusses this matter. I just can't find it right now.
I would encourage you to apply to every possible scholarship you are eligible for. If you are qualified to get in to USNA, then you are definitely qualified to receive scholarships! In many cases, the scholarship check is written directly to you.
Even though USNA is "free" there are still out of pocket expenses and some of the summer programs have a fee. Your parents will thank you over and over!