Thanks for posting all of the information!
I don't think anyone expects you to "get ahead" all at once by selling the car. They expect you to free up some money (because you will no longer have a car payment) to use for paying down your other debt. Are you in the city? Who is using the car, and for what purpose? Gett car service in Manhattan is pretty affordable and you could take 90 rides per month just for what you are paying for your car now. (
http://gett.com/nyc/) Of course, I don't know exactly where you are, and at $2200 a month rent with a family of 5 you are probably not in Manhattan. I'm just using that example to illustrate a point.
If you really need a car, then please ditch the giant car payment and get something older and more reliable with a much smaller used car loan. You can get a perfectly reasonable Honda or Toyota for $5K-6K (my used Prius was $3600 and it's been running like a top for 45K+ miles for me with nothing for expenses but gas, oil changes, and tires.) Also don't give any crap about "needing" a big car, because I'm a) probably taller than you and your wife and b) a former officer of my local parents of twins and triplets club; yet I drive a Prius. I found a twin stroller that would fit in my Corolla trunk back in the day.
You can then put what used to be the giant monthly car payment towards the credit card debt to kill it. I really think this will get you ahead the fastest.
I think your food expense is reasonable for 5 people in NY, although not everyone here will understand the prices where you live. It could always be lower, but even if you trim off $300 a month ($500 for 5 people would put it very low for NY), you still aren't making nearly as much headway as you will with a car switcho chango.
Also, excellent job on having life insurance, which is very important! Please don't let that lapse for any reason. Utilities are reasonable.
Phone is way too high, though, and I second looking at IP Daley's thread. I'll also second looking into lowering the student loan interest rate if you can do it in a way that makes sense.