Do I have any right to move out of state with my daughter without having to go back to court over this?

Relocation is a very sticky issue in Arizona. It requires notice to the other parent, who then has a right to object. I would suggest talking to an attorney about the requirements for relocation and whether you can proceed with moving because you have "full custody". The issue is more who has legal decision-making so an attorney may even want to see your original custody orders to help make that assessment.

It depends upon what is in your decree of divorce. If the decree of divorce is silent on the subject of whether you can relocate, or under what circumstances you can relocate, then there is nothing that would prevent you from relocating outside the state of Utah with a child. You must, however, provide the other parent with advance notice of your intent to relocate. The section of the Utah Code that applies is 30-3-37 30-3-37. Relocation. (1) For purposes of this section, "relocation" means moving 150 miles or more from the residence of the other parent. (2) The relocating parent shall provide 60 days advance written notice of the intended relocation to the other parent. The written notice of relocation shall contain statements affirming the following: (a) the parent-time provisions in Subsection (5) or a schedule approved by both parties will be followed; and (b) neither parent will interfere with the other's parental rights pursuant to court ordered parent-time arrangements, or the schedule approved by both parties. (3) The court shall, upon motion of any party or upon the court's own motion, schedule a hearing with notice to review the notice of relocation and parent-time schedule as provided in Section *30-3-35 * and make appropriate orders regarding the parent-time and costs for parent-time transportation. (4) In a hearing to review the notice of relocation, the court shall, in determining if the relocation of a custodial parent is in the best interest of the child, consider any other factors that the court considers relevant to the determination. If the court determines that relocation is not in the best interest of the child, and the custodial parent relocates, the court may order a change of custody. (5) If the court finds that the relocation is in the best interest of the child, the court shall determine the parent-time schedule and allocate the transportation costs that will be incurred for the child to visit the noncustodial parent. In making its determination, court shall consider: (a) the reason for the parent's relocation; (b) the additional costs or difficulty to both parents in exercising parent-time; (c) the economic resources of both parents; and (d) other factors the court considers necessary and relevant. (6) Unless otherwise ordered by the court, upon the relocation, as defined in Subsection (1), of one of the parties the following schedule shall be the minimum requirements for parent-time with a school-age child: (a) in years ending in an odd number, the child shall spend the following holidays with the noncustodial parent: (i) Thanksgiving holiday beginning Wednesday until Sunday; and (ii) Spring break, if applicable, beginning the last day of school before the holiday until the day before school resumes; (b) in years ending in an even number, the child shall spend the following holidays with the noncustodial parent: (i) the entire winter school break period; and (ii) the Fall school break beginning the last day of school before the holiday until the day before school resumes; (c) extended parent-time equal to 1/2 of the summer or off-track time for consecutive weeks. The children should be returned to the custodial home no later than seven days before school begins; however, this week shall be counted when determining the amount of parent-time to be divided between the parents for the summer or off-track period; and (d) one weekend per month, at the option and expense of the noncustodial parent. (7) The noncustodial parent's monthly weekend entitlement is subject to the following restrictions. (a) If the noncustodial parent has not designated a specific weekend for parent-time, the noncustodial parent shall receive the last weekend

Probably not, unless you can show that it is in your daughter's (not your) best interest.