The red zone doesn’t let you pick any game you want. It shows all Sunday games when they are within the 20 yard line, or the exciting plays if nothing else is going on. Sounds like you had a misunderstanding of what it covered.
It sounds like the package that you want is the 'NFL Sunday Ticket' which does allow you to view every NFL game in its entirety as it occurs live. The package is offered exclusively on DirecTV and not available on Dish.