La Quinta inns almost all are pet friendly, or at least dog friendly. Also there are some Motel 6's. On the higher end the W hotels all are, but there won't be too many of those along the route you are headed. You can check out http://www.dogfriendly.com/. It's how I'm driving with my two dogs from LA to Houston this winter.
How does this work at a hotel? Do they have specific rooms set aside for people with pets? If I stayed at a La Quinta, is it possible that a dog slept in the same bed the night before?
I believe it is possible. I could be wrong, though. When I stayed at La Quinta with my dogs before, there didn't seem to be any discernible way they made my room different from any of the others. There were a lot of guests there who also had dogs.
Absolutely disgusting. I do NOT want to stay in a room where a dog had slobbered all over the place the night before. Even worse, cats, because they make me sneeze , itch and have watery eyes. I actually made a conscious decision about a year ago to quit staying at the W hotel in dallas for this reason. I probably had spent 75+ nights there for work, and when I noticed a fair number of dogs in the place, I decided to stop giving them my business. Dogs do not belong inside a hotel under any circumstances.
I believe Holiday Inns have pet-friendly rooms as well. At least they used to a few years back. I can't say I know anything about Holiday Inn Express.
Maybe you don't belong inside the hotel. When I'm with my dogs they do belong in the hotel with me, as long as that's allowed.