I note if you are in fact admitted to the state university you described, and Rice was literally the only college on your list that you preferred, that would be fine as far as I am concerned. In fact many successful premeds went to their top in-state choice, because they are generally very well-suited for such students.
I guess the question is how sure are you about that state university? I would also think your HS counselors would be a top resource on that specific question.
If you are interested in a very, very good backup plan, colleges like Pitt and Minnesota are also highly recommendable for premeds (and actually lots of people), and they have rolling admissions. So you can apply and potentially get an answer even before you apply ED anywhere, and that can really make it easy to keep your application list short (assuming that makes sense for you generally).