When we took my little ones at that age (and even now, at 11 mos and 2 1/2), they would nap in the stroller. When it looked like they were getting pooped, I'd find a quiet, shady spot (yes, it IS possible! see below) and send DH and DD off to ride for a while, change the diaper, feed the bottle, and rock the baby in the stroller until she fell asleep. Then kick back and people watch while the baby naps.
As far as riding rides, if by "baby carrier" you mean something like a snugli, then you should be able to take the baby on rides in this. You might find, however, that your little one will be much more interested in the rides than you expect.
Whatever you do, take enough of everything with you. You do NOT want to run out of formula or diapers in the parks. While they are available for purchase, they are insanely expensive there.
Quiet places for the kids to rest:
MK - There are several nooks and crannies in Adventureland, near Pirates of the Carribean. Great to send the older ones for ONE MORE RIDE on Splash Mtn and Pirates!
AK - Anywhere on a bench in the Oasis area near the front of the park.
EPCOT - There are spots throughout the park, but the one that worked best for me was the walkway between the restaurant and the store near Mission Space. Not as quiet as some, but the constantly opening doors of the two buildings made it rather cool, and a relief from the opressive heat.
MGM - Not so many shady spots here. What did work was the covered food court area near Tower of Terror, across from the Beauty and the Beast show area. We hit it after the lunch rush, and it was quiet and covered.
Good luck to you, and have a super time!