We also moved from the States, and our kids are at SAIS. I don't know if they have space or not, but they do have kindergarten and preschool at the same school starting at age 3. My youngest, in KG1, is quite a stubborn kid, very difficult to have in a classroom, and her teacher has won her over. They are structured, but the younger ages have lots of free time and play time and I don't get a sense that there are too many rules to follow. My KG1 would have a lot of trouble if it were quite restrictive. One thing I liked about sending them to an American school is that they are still learning about our money, the American history sort of information, and they celebrate holidays we celebrate back home like Halloween, which is a big deal to my kids. It made for an easier transition for them, but kids are pretty adaptable so they probably would have been fine at other schools too.
We also used an orientations counselor, and I had the same quandry as you - how to pick a school I had never visited and therefore a neighborhood we would live in...it was quite confusing and difficult. I say pick the school first, stay in a
Serviced Apartment for a while until you get your bearings and then choose the neighborhood. Once you choose your school, the rest will fall into place. We used to walk to school also, but I really like the bus. It is just so convenient. The MRT is also an easy way to get the kids to school if the school is close to an MRT stop and you feel you want to do dropoff but not drive a car.
We initially talked our company into paying application fees for two schools, but eventually decided to go with our gut and only applied to the one we chose. If they don't have space, or have a long wait list, write them off your list and move on. You don't have time or energy to wait and worry, and those that do have space will be more than good enough. Most of the parents I have talked to are happy with their schools, regardless of the school. Trust your instincts, follow your gut, narrow down the list, talk to parents who have kids at the schools if you can. I even resorted to private messaging people on this forum when I figured out they had kids at a school I was considering, and people were very helpful. I bought the
International Schools guide someone else was hawking here, by the way, and found it enormously helpful. A small price to pay to figure the school situation out.