If your ex objects, the burden will be on you to establish that it is in the best interests of the children to relocate, not your own best interests.
Given your decree as written does not appear to give you carte blanche license to relocate against the wishes of the other parent, the court will most likely grant a request from the mother to enjoin the children from relocating until the dispute can be resolved in court.
This would throw the ball in your court to file a suit for modification of the order based upon a change in circumstances, although she could also file suit.
Most relocation cases in Texas give a lot of weight to the relocation imperatives enumerated in Lenz v Lenz. These include an improvement in the standard of living of the children, better job or educational opportunities for the custodial parent, and more. Bottom line is if your ex ojects to the move, you will likely have to frame your arguments to the court in this context.
Whatever issues of neglect your ex had, it apparently was not sufficient to deny her joint managing conservatorship of the children, as is often the case with issues of neglect. If you were sole managing conservator you would be in a stronger position than you are now.
If the main reason for relocation is to move in with a girlfriend in another state, with other reasons that are really only sugar coating for the main reason, then you might have a real challenge convincing a judge relocation is in the childrens best interests.