I'm not legally married. Am I in a sinful relationship?

For all intents and purposes, you are married already. While you may not have a legal marriage license as yet, a license from the government is merely a formality at this point. Both of you have obviously made a commitment to one another, brought children into the world and have every intention of continuing together in this relationship. Your commitment to one another is confirmed by the fact you would view any separation as equivalent to a divorce. 

Therefore, your situation is similar to the one Paul described in 1Cor 7:

1Cor. 7:10  But to the married I give instructions, not I, but the Lord, that the wife should not leave her husband
1Cor. 7:11 (but if she does leave, she must remain unmarried, or else be reconciled to her husband), and that the husband should not divorce his wife. 
1Cor. 7:12  But to the rest I say, not the Lord, that if any brother has a wife who is an unbeliever, and she consents to live with him, he must not divorce her. 
1Cor. 7:13 And a woman who has an unbelieving husband, and he consents to live with her, she must not send her husband away. 
1Cor. 7:14 For the unbelieving husband is sanctified through his wife, and the unbelieving wife is sanctified through her believing husband; for otherwise your children are unclean, but now they are holy. 

In this passage, Paul was addressing the question of whether a woman who comes to faith while married to an unbeliever should leave her unbelieving husband. Paul says specifically that as long as the unbelieving husband agrees to remain in the relationship, the woman is required to stay in the marriage. There is potential for her to witness in the family and bring her husband and children to faith, which is reason enough to remain in the marriage. In the meantime, a Christian wife and mother will serve to sanctify the family (i.e., provide a godly influence). 

Therefore, we believe you should remain married, and moreover, we strongly recommend you formalize your relationship by pursuing a legal marriage as a means of protecting your Christian witness within society. Though we believe the Lord views you as a married couple already, nevertheless many church congregations might consider a couple living together absent a legal marriage to be living in sin. Therefore if possible, we recommend you gently persuade your husband to obtain a marriage license. Regardless of his answer, remain with him and serve the Lord where you are.

You may find our 1Corinthians study helpful, especially the lessons on chapter 7.

Scripture quotations taken from the (NASB®) New American Standard Bible®, Copyright © 1995, 2020 by The Lockman Foundation. Used by permission. All rights reserved. www.lockman.org