As many people have said, cheap family-friendly lodgings are plentiful near interstate highway exits and other major roads. (In the US B&Bs tend to be pricier luxury accommodations; look for motels. Off-brand motels can be even cheaper but service levels can be spotty. You are unlikely to check into a chain motel in January and find that the heat doesn't work, for example.)
However, it is well worth getting reservations in advance -- even a few hours in advance -- as you might be surprised to learn what areas tend to sell out their lodgings either regularly or for some obscure local event. You will also usually get better prices booking online than by walking in the door.
If you have tents with you you can also consider some of the chain campgrounds (or other local campgrounds, but they are going to be harder to find online), but beware, in the US a "campground" is often primarily a place to park your RV for the night and may not be a friendly place to pitch a tent.