You can qualify for all kinds of assistance. WIC is a program that provided you with formula for babies up to age 12 months, and then until they are 5 years old the get milk, peanut butter, cheese, fruit, grains, etc...each month. There is also child care assistance. If you are that low income, even with working you will still get nearly free child care. You can also get medical cards for yourself and the kids so that everyone can have quality medical care.
In Oklahoma a family of 4 can get just over $2000 a month and still qualify for food stamps. If it is unearned income like SSDI or SS then it is figured a bit differently. For earned income they take a lot out before those people get their money, income tax, insurance, etc...so they get more assistance.
For low income housing it depends if it is section 8 or HUD or some other kind that your state may have. One program is where you live in their housing and the other is where you pick a home and the owner must agree and the house must pass inspections to be eligible. Ask the housing authority the questions and they should be able to answer them.
Another option is Habitat for Humanity. I was President of the Board of Directors here for some time. The program is for people who are trying and just need a hand up to be able to live in a better environment. Look online and you should find the nearest one. Call the number and get them to send you an application. Chances are that it will be a shorter wait too. You and your family will be required to work for about 350 hours over the period of time you are waiting to get your home. We had one mom who had severe asthma. She did mailers and fixed the snacks for the workers that were supplied to her. She did her hours off site in many ways. She took some budgeting classes, parenting classes, lots of things. Her dad worked every weekend and still to this day is an important part of the work crew on every house since theirs.
It is a wonderful way of getting a home that fits your family and that is something you worked for. The 350 "Sweat Equity" hours are your down payment. That applies to your home loan. Most of our home owners payments were under $200 a month. They still stay very cheap, most are still under $300. After the home is finished and you are moved in Habitat basically becomes your banker. If you ask you can even pay off the home early or discuss any issues you are having.
Habitat is a wonderful program that has helped millions of people worldwide to be home owners.