Congratulation TNMM!!! I'm so pleased this has come through for you!
My DH earns the same as your DH will (although he usually gets a bonus of around $4-8K as well). We pay $600/week rent. One Kid at public school (so negligable fees for us), and one at pre school 3 days a week, approx 40 weeks a year (=$120/week so, approx $5000/year).
We have 2 cars - one we own (be aware that new and 2nd hand cars are more expensive here than UK), and one we lease/buy on a Novated lease. Get your DH to ask his new employer if they do these - many big firms do. Its a way of paying the loan pre tax - so like salary sacrificing - and is very cost effective. I think ours costs us $600-700 per month - can't remember exactly, for a little mazda 2 that DH drives to work - was new when we got it.
I think you'll be ok on the amount you mention, but TBH we do stress a bit about it at the moment. We always save DH's bonus and any tax we get back etc, and so manage to have about 15K in savings, but we are very often drawing down on this for month to month expenses, so its doesn't really get saved yswim. Not having DH's car would help a lot with cutting costs but I'm not willing to spare him for the 4 hours a day it would take him to commute!
We always have enought o eat and wear, and the kids do swimming lessons and activities, but we don't have spare for eating out, shopping sprees or anything like that, and try to budget fairly carefully for food, make cupcakes for school lunches instead of buying anything pre packaged, fill drink bottles rather than buy packaged drinks etc. We are by no means on the bread line, but can't spend with abandon either.
Oh, also DH is paying off student load still, at rate of about $650/month, so thats an expense you wont have.
I do hope that helps with your planning.