1) How bad is the roof? Do you know anyone who might be able to pitch in for a DIY job? Dad, brother, sister, friend, cousin - church/community outreach type group. If you need to shell out $6-8K for this thing, look into every possible way to reduce that cost. If you can pay for only materials and some pizzas/beer/soda you will cut your costs in half. Savings - maybe $3-4K.
2) Student loan - I wouldn't pay anything extra if it might be forgiven.
3) You need to sell your car. You have no real choice here unless you want to hopefully save $80/month in your budget. You net about $31K/year, and you have a $15K car loan. This is not ok, not at all. You are likely upside down on the car, but I have no idea since you didn't give any details as to what it is. Take all the savings from your DIY roof job and use it to pay off the difference between your car value and car loan. Savings - $350/month.
4) Buy the cheapest car you can find, as long as it has seat belts for you and your 2 kids, and it will get you to where you're going somewhat reliably. Pay cash if you have any left. Get a loan however you can if you have to, but you shouldn't have to because I'm talking $1,000 on Craigslist, not $7,000 at a dealership.
5) Carry only the car insurance you are legally required to have. If you have a $1K car, you don't need any extra coverage on it. Savings - maybe $75/month.
6) Food - You need to read this if you haven't already
http://www.mrmoneymustache.com/2012/03/29/killing-your-1000-grocery-bill/ . You are obviously not spending $1,000/month on groceries, but it will give you some ideas. Here's my main thought on the food - eat cheap stuff that's nutritious enough, and never, ever eat more than you need, or throw any food out. Don't go to the grocery store when you have leftovers, or when you have edible items that can be turned into a meal.
7) Swallow your pride and take advantage of whatever social safety nets might be available to you. Food stamps, WIC, earned income credit on your tax return - the last one could allow you to reduce your tax withholdings which would increase your net pay. If you have a tax refund coming, use it to make this car transition happen. It's quite possibly waaaay more important than your roof.
You are correct, you do live a fairly frugal life, except for the car which is absolutely killing you.
If you make as many of these changes as possible things will look a lot better. Then when the daycare/diaper costs disappear you will have plenty of extra funds. When that happens, don't be tempted to pamper the kids with too many expensive activities. Parks/libraries are free.
Good luck!
