Regarding your interest in both engineering and business, GTech has a very solid business school itself and an Engineering and Business minor.
With your excellent academics and ECs, you have a reasonable shot at merit, although by no means guaranteed. When it comes to merit, GTech places the most emphasis on service and leadership. They want kids who will contribute to campus life and continue to contribute to society at large post graduation. So I see that as a reasonable application for both your interests and the potential to meet your budget.
Edit: Oops. Posted before I meant to… I agree you are lucky to be a VA resident since you have such strong in-state options. But a quick look at the UVA and VATech results on CC show that you need a more likely school on your list just in case. Whether that is in-state or an OOS with a good chance of merit, please take time to find a couple of more likely schools where you could be happy and will meet your budget. Once you have that, I encourage you to take a shot at more prestigious schools and competitive merit.