Somehow I suspect if the OP asked the sales department for 10 or more rooms (as opposed to asking the front desk for 1 room), there would suddenly be availability!

I remember calling some hotels, and sometimes when the receptionist is too busy to answer, the call would go to a recording that would say something like "Thank you for calling xxxx hotel...for individual reservations of up to 9 rooms, please press 1 [you would then be transferred to central reservations], for group reservations of 10 or more rooms, please press 2 [you would then get transferred to sales]..."

I might have also recall seeing a walk-in guest at a hotel get handed the telephone after the front desk staff presumably called central reservations to make the reservation first before they can be checked in.