No, that is not totally true (6 month time frame) and here is why. In NY,
child support goes to at least 21 so what would stop every custodial parent from moving here and getting it changed to get 3 more years of support...?
You can not just ask the new state to make an order but they can enforce an order from another state.
Since she is there and sounds like she may be involved with social service, the case prob will stay there.
Get ducks in a row. Courts like to have kids with their mothers still. Prove she is doing well in school, have your lawyer stall a few times if possible to have her in school longer. Bring
report cards and attendance records,
medical records of a healthy, up to date child. Show you are willing to have mom be apart of her life as much as possible as long as she is clean/sober/whatever the issues are. They give custody to the parent who will make sure child sees and is apart of both parents/families.
Do you have temporary custody now or perm custody?
It is not usual for a parent to pay child support when child is not with them more time. It is to make sure child is taken care of in both households and close to the same standards for both (or at least that is the reasoning behind it...that if you were together, child would have XYZ and still should in both houses).