January is not too busy so the lines shouldn’t be too long and you shouldn’t need fast passes. I visit often living only 15 minutes from the park and my 6 yo has no problem walking around on short visits. I do bring our stroller when I know we will be there late at night or for long hours (Halloween party and Christmas party). It is much easier to have her sit in there while going through the crowds to get out of the park.
You can rent the double stroller there, it will hold both of your children and I like having my stroller to hold drinks, my cooler, etc. All the stroller parking areas have employees to keep the strollers organized so your stuff will be fine if left in it (there are also cameras EVERYWHERE). If you do rent one you have to get one each day and you cannot bring it back to the hotel.
I would use a stroller at Magic Kingdom, Epcot and Animal Kingdom, Hollywood Studios is not that big.