No, they do not have an obligation if you do not have a reservation.

When I worked front desk, I routinely had people who "forgot" to check out, and then when reminded "forgot" the time...
I had assistance from the local constabulary to accompany these people from the building.

(This was a smaller property, so it was out policy that if someone left after 1pm to charge them for the full room rate the extra night, as their late departure time would cause significant disruption to our housekeepers.)

You might be lucky - people often cancel rooms, so some may become available. Assuming that everything on the net IS booked and has been for a little while, they may already have a waitlist, or some non-internet sell rooms available. You could call and ask. Tell them you'll take a twin or single room and you might be lucky