score:4
In general, no, you don't have to, particularly if you're just looking for a bed for one night, and aren't terribly picky.
That said, the later you choose to pull off the road - particularly in peak travel season - the more likely you are to be greeted by that dreaded "No Vacancy" sign. (Or, in nicer establishments, a polite refusal at the front desk.)
My typical MO when driving long distances, is to pull off the road at a rest stop or town about 2-3 hours before I intend to stop driving for the day; (in my case, this is usually right around the time the sun is getting low enough in the sky to blind me if I'm driving west, so I can wait it out; otherwise, around Sunset.) Then, I'll use my Smartphone (or, if I'm running low tech for some reason, a payphone at a rest stop and a map), to look up hotels that are about ~100 miles further along the road from where I am, and book a room online before continuing my drive. This gives me the flexibility to book same day for wherever I end up, while avoiding the occasional frustration of showing up at 9 PM to no rooms for 50 miles in any direction on a Friday night, followed by sleeping in the car in a parking lot (it's happened to me. It sucks.). All of the major US chains have pretty decent mobile sites that are geared towards exactly this sort of use. I find the IHG (Holiday Inn et al), and Best Western apps are particularly good for looking a little bit ahead along a highway. YMMV.
Keep in mind that in some cases, booking further in advance can give you access to better pricing or promotions, but you give up flexibility to do so.