The stroller is a very helpful thing to take with you, instead of renting one while you are there. It helped us a lot in the airport, since it is so big we just strolled him right up to the door of the plane and then they give you a red tag to put onto your stroller and one for you to keep to claim your stroller after you land.
It is also helpful to have for your resorts. I've never been to Swan & Dolphin before, but if it is like most...you will have to walk a ways to get to your room and to where the bus picks you up.
Then once you get off of the bus (or if you rented a car) it helps to have the stroller there to help you out. Especially if the child falls asleep when you are ready to leave the parks. You can leave the child sleep in the stroller until the bus arrives to take you back to your hotel...sometimes this can be awhile.
If you bring a stroller that reclines that is also helful, so that way your child can sleep while at the parks and you don't feel like you ahve to head to the hotel everytime the child is tired.
The have baby care centers at each of the parks, so when you or the child need a quiet get a way, this is a perfect place for you to go and relax.
Good luck and if you have anymore ???, just ask away.