If you do book two separate reservations, then it should be no problem for you to ride DME to the airport with your children then back again. For the ride to the airport, give them the flight # that your children are on for all of you. You all board DME, your children's bags can be checked with RAC if on an airline that participates or they can be transported on the bus and checked at the airport. Give your bags to bell services and they will be moved to the new resort.
For your return to the new resort, give the reservations folks "any" flight number that arrives about the time you think you'll want to board the DME bus. After you get your children delivered safely to their gate, grandma, or whatever, then walk back down to DME, give them your boarding passes for the new reservation, then you will be directed to a bus. You will be taken to your new resort where you can check in, then begin your adult getaway.