Hello,
Good luck.
The first thing I have to say is that having 2 or 3 children to register is not always a hinderance. School are as happy as any business to take on 3 deposits & fees in one go. Its not cynical, that's just the way it is. So don't let that scare you.
However you are applying for the most coveted early year groups, so don't delay, start applying today to give you a bit on precedence on the waiting lists.
Briefly on the top 2:
Tanglin now has a debenture system but as you mentioned you are not on an expat package, forking out debentures for 3 kids is not an option I guess?
UWC has also changed their wait list system last year. The lists no longer roll over for years and the East campus is fully operational. Check on their website for updates, they actually do tell you for with year groups they still have vacancies.
Its definitely worth contacting them too, they might have a place in a certain year group you are applying for and then that will give sibling priority to your other children.
Many other good schools. SJI, canadian, Dover, SAS, stamford, Australian, Avondale, OFS to name a few.
Do you have any hang ups about changing systems?
Most International schools run a full IB program me and all the above have consistently sound academic results.
Just call all of them/email and start applying. The application costs are frankly something you should ask the employer to cover as for 3 kids that will add up.
Don't be stressed about talking to the schools. Remember they do it all day long, its their job and its only an emotional issue for you, not them 
The job may come with a relocation agent. Some are very good at navigating the school system but they will just point you in the right direction. They can't "shake things up" and find you a place. Unless you pay for a debenture that is.
The children may end up at different schools for the first year but school buses run very efficiently so don't stress out too much about it.
What year group are they if you don't mind my asking?