Could we (as a couple) stay together in a hotel in Dubai?
Yes you could be able to stay at a hotel in Dubai as an unmarried couple. Women in Dubai often keep their last names.
Islamic law does prohibit unmarried people from living with each other which does include in a hotel room.
Try to be discreet that you are only a couple (unmarried) in Dubai. Refrain from calling the other person your girlfriend or boyfriend and also fiance. So it is not legal, but most of the time they don't act upon it.