I'd use two single umbrella strollers. Buy them at wallmart or ToysRus before the trip they'll be very inexpensive - consider a good investemnt for your trip - they'll be good in the airport too. If you can't carry them on the plane they will stow them with luggage at the time you board. Even if the kids want to walk some, they will get very tired quickly. Plus you always have to drag stuff around for kids so having strollers to carry stuff is helpful too. You are allowed to bring certain types of snacks in with you - & water so having a tote bag or packpack on the strollers is helpful. Two single umbrella strollers will allow you to be so much more nimble than a double or a large jogging stroller (which is never a good idea for an theme park no matter what - they're just too big). consdier also when one needs the bathroom now (how far away is it?) while the other is napping, or one wants a trip to see the princess while your husband and older child want to go on the teacups. Two small umbrella strollers offer so much more flexibility and bought at walmart will cost a fraction of renting - and you get to keep them.
As for your dad - you can rent motorized scooters outside of Disney (they will deliver it to your hotel and pick it up later) for about the same $ as a wheelchair in Disney. My husband had ruptured his achilles tendon last year about 6 weeks before our trip and was 1 month post-surgical when we went. The scooter was a God-send - but it was not sufficient to also carry a preschooler on it with an adult. The basket in front was handy - we used it as a "cooler" of sorts - double plastic bag as a liner of the basket and doubled up ziploc bags of ice to keep drinks, spray bottles, etc. cool.
You can find scooter rental information by googling scooter rental disneyworld - there are about 6-8 companies -all are similar prices. Ours began having ignition problems and they brought us a new one in about an hour's time. All the buses and boats at Disney are accessible with the scooters & wheelchairs too.
Final note - the scooter/wheelchair will no longer get you to the front of the lines all the time. In some instances - on the older rides - it did since they don't have the queues set up wide enough to accomodate the chairs. But all the newer rides are accessible & the wheelchair / scooter fits easily so you won't get special accomodations on those. In some isntances - like the 3-D shows it ends up being worse to have a scooter/wheelchair since you'll be automatically seated in the back corners - where the 3-D effects are not nearly as good. If your dad can easily transfer to a seat speak up early when you get to the 3-D shows.
Good luck mama - it's always an adventure - and you'll have a good time no matter what - but being able to move about, to take one kid one way, and another another way in separate strollers may help too. Have fun!