Texas is a common law state.  
How have you defined your relationship?  Do you consider yourselves as married and are husband and wife?

How have you been filing for the last 8 years? Are the children the biological or adopted children of your partner?

You may be able to file as Married Filing Jointly if you live in a state that recognizes common law marriage. Currently, the following jurisdictions recognize common law marriage:

 Alabama (if entered into before 1-1-2017)

 Colorado

 District of Columbia

 Iowa

 Kansas

 Montana

 New Hampshire

 Oklahoma

 Rhode Island

 South Carolina

 Texas

 Utah

Alabama (2017), Georgia(1997), Idaho(1996), Ohio(Oct. 1991), and Pennsylvania(2005) are grandfathered for the marriages before the year indicated. Living together in a common law state is usually insufficient - you need to hold yourself out as married including owning property together, having joint bank accounts, etc. To find out your state’s rules see:

http://family.findlaw.com/marriage/common-law-marriage-states.html

This comment repeats most of what Opus17 has already stated in his answer. Is there some part of it that you don't understand?  Have you followed the 3 parts in Opus17's answer that are required for you to be common law married?

If you have followed them and consider yourself common law married, then you must file either married filing jointly or married filing separately. It is almost always better to file jointly than separately. Texas is a community property state which makes filing a married filing separately tax return somewhat complicated. You are not required to submit the document for informal marriage to the county clerk to be considered common law married.

If you have not followed the 3 parts to be common law married, then you each must file single or if eligible, one of you could file as head of household and the other as single.

Once you are married (common law or otherwise), you must adhere to all the laws for married couples including community property laws. You must get a legal divorce if you choose to no longer be married.

@cindyarboleda20 there is no form to fill out for informal marriage.  The whole point is that it happens without paperwork.  It happens by 2 private actions and one public action,
1. you agree to be married (private)
2. you live togehter as if you are married (private)
3. you tell at least one other person that you are married (public)

If you have filed a married tax return in the past, that will hit all three buttons, since you can't file a married return without agreeing to be married.  That means you are married whether you want to be or not.

If you have not filed a married tax return before, then it sounds like you aren't married because you haven't agreed to be married (when you said "and I will be honest most likely plan not to.")  Then if you are the person who pays more than half the living expenses and you have at least one qualifying child dependent you can claim, you file as head of household.  You can't claim your partner as a dependent if he earned more than $4050, so he would file a separate single tax return.  You need to claim at least one dependent to qualify as head of household but either one of you can claim the second child, if they are both your children.  Just make sure that each child is only claimed by one of you.  In turbotax, the parent who is NOT claiming the child should not even list them in the program, due to potential for confusion.

There's no special reason that claiming him as a dependent one year and not the next year would raise a red flag--it happens all the time.  And even if it does the truth is your best defense.  The only red flag would be if you filed a married return in the past, which would be a public declaration that you agree to be married.  Then, you are stuck with that until you get a real divorce.