I think you should take a tour bus. They can go places you are not allowed to drive and the shuttle will get you around the park too. May can have snow near the access point to the park or at higher elevations. I was just there a few days ago, lots of cars in the valley area and road work. Check with the park in advance to see which roads may be closed.
You can do either. Depends on your liking, driving yourself is a convenient option too, since you have a lot of time at hand, you can do your research about the points that might interest you and drive to the beginning of the trails/treks. Parking Is free at all points
We attempted to drive ourselves and then park at the visitor center, it was extremely crowded and took 3 attempts to find a parking spot (we went in Oct). I HIGHLY recommend you use the shuttle. They stop at every stop and then you don't have to worry about parking.
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