I concur with my learned friend's comments above - I'd say that you should be fine provided you 'cut your coat according to your cloth' as the old expression goes.
There are people who come here on packages 2x or 3x yours, spend all their money and then leave as broke as they were on arrival. Equally there are many families on far lower packages (probably the majority of 'expats' if you take the term in its widest sense) who manage to have an enjoyable life here
and save up a sizeable sum in the process.
We've never paid more than $5k per month for any of the (3 or 4 bedroom) places we've lived in here; in fact we're paying significantly less than that right now for a decent 3 bed apartment in a nice
condo in Tanjong Rhu. As JR8 implies, by being flexible and eschewing the idea that you 'have to' live in one of the most fashionable and expensive districts, you can find some decent places at (relatively) attractive prices.
The standard piece of advice we tend to give out to newcomers here is to stay in a
Serviced Apartment (ideally paid for by your employer) for the first 1-2 months; that will give you time to get to know the various areas and then hopefully find something reasonable to your liking. Rushing into a rental contract on arrival can be lead to a mistake that you regret for the next couple of years while you wait for it to expire.
As regards schools, with your children's ages you could of course look into the local school system. As you can imagine, there have been many discussions on this in the past, so a search will bring up various threads on the pros & cons, practicalities, etc.
If you do want to go down the
International School route (which is what we did), then you need to get their names on the waiting lists for the appropriate entry years as soon as you can. Most (but maybe not all)
International Schools tend to be over-subscribed, which is why you'd probably need to budget around $30k per child per year for one of the most 'popular' ones. Again, this is a topic that is much discussed on this and other forums (sometimes with a fair amount of bitchiness thrown in unfortunately), so a liberal use of the search function above is highly recommended.
Good luck for the move and don't worry - you'll be OK!
Edit to add: Regarding a car, I can understand the attraction of this considering you have young children, but you might want to try without one for a while - you'll probably save a small fortune in the process, even if you use a lot of taxis. We've never had a car here as I just think the cost is outrageous - I'd much rather save and/or spend the money on other things. Plus it's better all round for my health - see the comments from me, SMS, et al in
this thread for context.
We've enrolled both kids in a nursery over the summer and they're both enrolled at OFS for August (to save on costs, thinking to keep younger in the nursery for another year) 
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