You have two possible scenarios here.
One: You might try to prove that you had a common law marriage. In order to have a common law marriage in Texas, you have to do the following. 1) Agree to be married. 2) Tell others you are married. 3)Live together. It sounds like right now you may meet two out of the three, but you have to meet all three. A Judge would have to agree that there was a common law marriage.
If you can prove a common law marriage, then you would have to file for divorce to exercise your rights as a spouse.
If your boyfriend bought the home before your common law marriage started, then you would not have any rights to stay in the home.
If a common law marriage is found, you would have a right to a just and right division of any property that was purchased or income that was earned during your marriage. Since the children are not his, he would not be required to pay any child support for them. Since you've only been together for 6 years, even if a common law marriage was found, you would not get any spousal support because Texas requires that the marriage be at least 10 years long before spousal support is granted unless the paying spouse was convicted of family violence against the non-paying spouse.
The second scenario would be if a Judge did not find that there was a common law marriage. Texas does not have the concept of palimony, or support after a long term relationship ends. So, if your boyfriend owns the home, he may evict you and your children legally. He would not have to support you. You would only be able to take what property was yours and what money was yours that you earned in the joint checking accounts.
I do not know that you meet the criteria for a common law marriage. I would sit down with a lawyer in your area (many offer free consultations) and lay out all your evidence that you have to argue for a common law marriage and see what they think.