Yes, if you get any kind of scholarship money you still have to apply to school just like anyone else. The difference is you'll likely have lots of help from the coaches and staff to make the process easier.
However, if you think you're going to get a 'full track scholarship' you're either in the top 1-2% of runners in your state or you don't understand that track scholarships are limited and very, very few kids get a full ride in the good programs. I think it's 12.6 spots for men and 18 for women. Coaches have to spread those spots out among T&F distance, cross country and sprint, jump and field positions.
Also, it depends on the school, what the coach needs that year, who he signed last year, your grades (coaches like to have kids who will make their team GPA numbers look good), who on scholarship is graduating and freeing up a spot, who else is your HS competition for that same scholarship, etc.
For example, I know of a girl who ran 5:15 for 1600 and could only get an offer of 25% of tuition from a particular school, but two years later that coach signed a girl to full tuition scholarship with a 5:19 PR. It's different from year to year. Lots of factors went into that situation.
What are your times? Are you male or female? Maybe someone here can give you an objective idea of what you might get at certain types of programs.