Based on what you have written, it appears that you may have a claim to an informal (common-law) marriage. If you and the boys are physically safe, first speak with a lawyer, before leaving. There are ways to hire a lawyer, and you may qualify for indigent or low-cost representation. Even if you are not married, you are still entitled to support for the boys. It is important to not make any changes in your lifestyle until you are able to sit down and speak with a lawyer.
You desperately need to talk to an attorney.
If you are living in the home, then you cannot be removed until a judge orders you to move.
Even if he owns the house, if you have been living there for less than 10 years, he cannot ask you to move. You will need proof (driver's license, utility bill or a letter) addressed to that address.
You might have a common law marriage but you need to talk to an attorney about it.
Of course, you are free to take both children and leave.
Again, I urge you to talk to an attorney immediately.
I don't know where Overton is located.
I only practice law in Harris County, Texas.
The other attorney that answered your question is a great attorney and perhaps she lives in your area.
If you are not safe, then take the children and leave. I encourage you to have a safety plan and go to a shelter if you have no where else to go.