DoD Password Complexity: Users cannot reuse any of their previous X passwords

The character distance requirement as stated is only for the (one) previous password, not the 10 previous. Assuming your password tool requires entering the current password as well as a new one, you just check against that; no need to store anything there. (Also noted on this answer to the post you mentioned.)

The requirement of not matching any of the previous 10 passwords, of course, is handled by just checking against the old hashes.