The eternal stroller question!!! I would definately bring your own -- it is worth the agony of lugging it through the airport. Actually, folding it down for security checkpoints is the worst. Once you get through security, you have a way to keep your kids contained and also a luggage cart if they are walking!! You can gate check your stroller and it will be waiting for you when you deplane. Plus, you will be glad you have one the minute you get to the parks, instead of waiting in another line to payout more money! It is also handy for around the resort and DTD.
If you don't want to deal with your double, you can always use 2 lightweight umbrellas. I would be leary about the park rentals -- they are hard plastic and I think a double is $18 a day (check out stroller info at allearsnet site - I am too new to post link).
I would also think about the sit n stand. I have found them hard to steer, heavy and both kids may be fighting for the comfy seat, especially during a long, hot day. Your regular double may be the best solution, so each child has a familiar, comfy, reclining seat (you may be lucky and even get a few naps). I think the sit n stand and regular double stroller would be equally heavy/awkward to lug around, but the regular double would at least be more comfy for each child for long, hot days.
If you look at the cost of park rentals, you are almost purchasing a new stroller anyway. The Maclarens are as fabulous as everyone says, hold enough weight to make them viable for older kids and fold up and carry very compactly. Lugging a regular double stroller on and off busses is no fun, but immediately forgotten about once the destination is reached.
Sorry this is so long..... I guess I have a lot to say about strollers. Good luck with your decision.