Chiming in to agree with everyone else.....
yes, it's a holiday...and that makes it VERY busy everywhere
If you don't make a reservation now you will pay RACK RATE....that's the highest that they think they can charge. And it will be even HIGHER because it's considered a holiday. In other words...the $80 per night room that you could book now...will be $200 when you walk in off the street.
I applaud your financial responsibility....though it does make days like this a bit more difficult, it's not impossible. The other poster is from Canada and told you how to get a pre-paid credit card....I strongly urge you to do this. These cards are available "everywhere" here in the USA, including convenience store/gas stations like Circle K.....I'm not from Canada so don't know where else you can get one, but they've gotta be around, lol. Also, check with your bank....very often they offer things like this, though it may not be advertised widely, or you may have missed it since you are staying away from credit cards.
As a last resort....contact the hotel of your choice (call the actual hotel, not the chain headquarters or reservations) and ask them if you could send along a check or money order as security since you don't use credit cards. By calling the hotel itself you'll deal with a person who wants you in their facility and might be willing to bend the rules a little, rather than a reservation clerk in another state who doesn't want to go through those hassles, because they have a quota of calls to answer. I've gotten many different special requests handled by the local manager when the reservations people have told me no. Including allowing a 17 year old to check in by himself (he was driving to stay with his aunt for the summer and hotels require you to be over 18).
Another thought...if you have family who has a credit card, would they be willing to book the guarantee with the card for you? Most hotels don't charge your room until you arrive (it will say on their site)....so when you check in you simply tell them that you will be paying cash, not using the card. They may require a deposit to cover any "incidentals" like phone or room service, you can either pay that, or tell them to shut the phone off (yes, hotels have that ability, they just don't like to use it, because someone has to remember to turn it back on when you leave or the next guest will be mad when it doesn't work, lol).
If you wait you'll pay more....you're likely to have to try a few hotels (or the lower rung of hotels) before you find one with availablity....and after 19 hours in the car, wouldn't it be nice to have enjoyed the ride, not spent all 19 hours worrying about how you'll find a room?!! For me, that would be a tiring ride as it was, I wouldn't want any worries to add to it....I'd rather be thinking about how much fun we're going to have and what to go on first....those are the max worries I'd want!