Although the "Choice of Managing Conservator" document is very persuasive to the court, it is not binding, as the court will attempt to make a decision which is in the child's best interest (which is not always what the child wants).
There are two ways people can form a common law marriage in Texas: If you are married pursuant to common law and no longer wish to be married to your common law spouse, you will need a divorce.
This document spells out who gets what property, where the primary residence of the children will be, how much child support will be paid, and how various child-rearing decisions will be made in the future.
Whether there is a settlement agreement or a trial, at the conclusion of the case a Final Decree of Divorce is drafted.
You are "married" until a court enters a final decree.