Agree. Just submit it. You have done your bit by asking them and chasing & it now, is not your problem.
Besides it's not as if the book goes into print at 9am tomorrow. The publishers will have allowed time for editing, typos, changes + legalities. They'll be used to these sorts of hicups. Even though you have been given a deadline, there is probably leverage for such delays. Try not to stress.
All the best with it.