Head north on the island (don't worry, there is only one main road) you'll soon be able to drive no more. Here is where fine white sand covers the road and creates this man-made barrier that separates the north and south padre islands.
Its a surreal sight to behold, with soft undulating dunes on both sides of the road. Getting out and climbing the dunes comes naturally, it's pristine because you've effectively left those spring break partying crowds far far behind. Here is where you have the sun, sand, beach all to yourself.