I do think that if he does put himself up on the block (Adam), then it would be Sharon or James as HOH. I say this because when they were couples and could not come to a decision, it was the 2nd place HOH winner that became HOH.
Natalie thinks that if Adam put himself on the block, she would become HOH

OMG could it happen 
bb wont let adam step down as hoh, and put himself on the block, now adam could go to the DR and say he wants to leave, and bb would have to let him, but no adam cannot put himself on the block, he can only walk out the door, if he wants to go on his own!
are you sure? I could have sworn thAT i heard something about reading it in BB rule book and they can do that?
100% sure, adam can leave the house go to the DR and say bye bye. but he cant put himself on the block and let the HG vote for him or agaisnt him, i have watched bb from day one 1st season, hoh can leave but cant put themselves on the block, why> cuz say adam and the others want someone OUT., and the only way to get that person out is to put up adam, well adam is hoh he is safe you cant use adam on the block, but adam could DUR,,leave on his own! i am 1000000000000000000 % sure of this!