If you want to bone up on extra sciences, every little bit helps. I have a couple biological sciences degrees, and have found that to be very helpful in my nursing and NP studies. I’m not familiar with what a “medical sciences” degree covers, but I’ve always had the impression that they were designed to appeal to folks that wanted to provide a boatload of biology/science credits to help folks increase their low GPAs after getting rejected from some sort of professional school (at least that’s how I’ve seen people try to use them). The schools that offer them are certainly excited to have your money (they are expensive), but I’m not sure if they are geared towards adding robust knowledge to pre NPs. I think I’d just take hard sciences as electives where you already go to school.
Well over a hundred thousand NPs prescribe without physician oversight. If you want to have a net underneath you, you can probably find a physician or organization that will hire you and stick close. Or you can locate to a state that requires it, although each year at least one new state opts for independent practice. You could also become a nurse and then go to PA school, which I think would be a good option for you. Just take classes that are the prerequisites for Pa school instead of throwing away money on the medical sciences degree. Then throw your money at a PA school, and you will always have the oversight you crave, as well as training that you may find to be adequate to assuage your fears.
So to answer your question... go be a PA. It’s an obvious solution.