There is a presumption that parents should have custody of their children which is possible but difficult to rebut. If the grandparents have been taking care of your daughter for a long time, they might pursue third-party custody and guardianship. To win they would need to prove some very negative facts about you. It is likely that you will continue to have full custody of your daughter if you are a fit parent and you are taking care of your daughter now.
If paternity was established, his parents can petition for grandparent visitation rights. The court may give his parents the right to spend time with the child if it is in the child's best interest. That's a subjective standard. If the grandparents have spent a lot of time taking care of your daughter it may be in her interests to maintain those relationships.
If you need to move 1500 miles away for a good job that will make it possible for you to provide for your daughter, it may also be in her best interest to move.
It may be possible for the court to allow you to move and also protect your daughter's relationship with her grandparents (for example, even if you need to move very far away the grandparents could visit your daughter a couple times a year for holidays and/or have weekly Skype/phone conversations with her.)
If the grandparents file a petition for visitation, guardianship, or third party custody you need to be prepared to protect yourself from any negative allegations and explain how moving 1500 miles away will benefit your child. A lawyer who has access to more facts about your case- even a short initial phone conversation- would be able to give you better, more detailed advice.
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