your options are
Do a short sale if there's enough time
Start making moving arrangements
After the sheriff sale you have a 10 day right of recission - this means you have an extra 10 days to either sell & close on the property or come up with the full amount owed the bank or move .
Don't let people tell you they can help you and then they start charging you a monthly fee for their service - I have a few accounts with banks and I always come across home owners that have been foreclosed on and the tell me about these crooks that the were paying to help them with the whole foreclosure issue.
Bottom line is either you find the money to pay the bank or you vacate the property , the age of your kids have no effect on the outcome. If it was winter the judge might give you a brake ( extra 30 days) , but it's summer now and schools are out , so your out of luck.
Almost forgot , if you wait till the day of the sale you can request an extension which will give you an extra 2 weeks , I think our allowed 2 requests . My advice is to just start seeking alternative housing immediately.