Keep in mind that since B6 is a revenue-based program, there is no concept of "award" inventory per se. Assuming it's JetBlue inventory, then if paid seats are available then "award" (i.e. points) seats should also be available. The reverse also holds true.

If the B6 BFF is telling you that seats are available, but you can find them (paid or points), then the BFF is probably broken and returning phantom inventory.