Always book directly through the hotel website. Priceline has deals but their customer service is horrible. Also it seemed like the majority of the time I would book an express deal they would put me in an accessible room. I’m not disabled and generally no issues with it but I found it odd. Not only that, they will never honor their “price match guarantee.” They state that if you find a cheaper price on a express deal they will refund you up to 200% of the difference. They will act like you are crazy if you call them after finding a cheaper rate. I learned my lesson the hard way. Don’t book through 3rd party travel websites.
Theory: people who need the disabled room are likely aware of this and are more likely to book ahead. Therefore, when you get to day-of or very close to it, if the disabled room is still available, it's unlikely it's suddenly going to be reserved by an actual disabled guest, so they may as well offer it cheap if it's more profitable than letting it sit empty.