Are we morally responsible for what we do?

Yes. Responsibility rests on the will that produced an act and on the record of the act, and it needs no proof that some other act was possible. Harry Frankfurt showed this in 1969 with a thought experiment. Black wants Jones to do a certain thing and stands ready to force him if Jones shows any sign of choosing otherwise. Jones does it on his own, and Black never moves. Jones could not have done otherwise, and he is responsible anyway, because the act came out of his own deliberation. Whether some other setting of the world was reachable is a question about the space of possible records. Which entry got written, out of whose repair policy, is a question about this record, and this record is what everything downstream is computed from.