I just looked up what a $280K mortgage (that's the $350K house with a 20% down payment) would be at current rates--$1,364/month. Around here the property taxes on a $350K house would be close to $12K/year, so another $1K/month = $2,364/month without house insurance, utilities, etc. That's nearly half the take-home pay. Add in a couple of kids, and this is quite a bunch of money.
It sure is, but I assure you, it's do able. Until very recently, we were paying 50% of our take home in rent alone. Utilities and bills were another $1200/month. We have 2 kids and live in southern CA.
Where the heck do you live that property taxes are 3.5%? Also, though, taxes are based on the assesed value of the house (land + "improvements") and not on the actual price/value of the house itself. My sister's house is worth $1.5M but her taxes are on an assessed value of $850,000. There is no way a $350,000 house will pay $12,000 in taxes.