Three other ships is normal. Unfortunately, you are correct that the old town is too far to walk with young children. I am guessing that for one day a car rental does not make sense. Does Disney offer a shuttle to old town? FYI, Old San Juan is hilly. Not bad for adults, however, might be a bit much for the 19-month-old and maybe even the 5-year-old.
Personally, we like the forts in Old San Juan, they take the national park pass if you have one.
When do you dock, when do you leave?