FAFSA is completely separate. The tax code is its own thing separate from everything else, and you follow the tax laws on your tax return, no matter what you said on the FAFSA.
You can claim your child as a dependent if your child,
a. lives in your home more than half the year
b. does not provide more than half their own support
When your child turns 19, they must also be a full time student (for at least part of 5 months) for you to claim them as a dependent.
If your child lives in your ex's home more than half the year, your ex can claim the child as a dependent.
In order to alternate years properly, the custodial parent (where the child lives more than half the year) must give the other parent a signed form 8332. That allows the non-custodial parent to claim the child tax credit but not head of household status, EIC or certain other credits, because those tax benefits always stay with the custodial parent and can't be waived, shared or transferred. If you have just been trading the dependent by claiming the child lived with you when they did not, the non-custodial parent may have been claiming improper tax benefits and the custodial parent may have been giving up certain tax benefits.
HOWEVER,
Note that when the child turns 18 (or reaches the age of adulthood in your state if it is different from 18), the special rules for children of divorced or separated parents no longer apply. You then go back to the ordinary rules which says the child can only be claimed by the parent where the child physically lives more than half the year. The dependent claim can't be alternated after they turn 18, unless the child's living arrangements actually change as well.
THEN,
regarding the child's tax return: If the child has income and files a tax return, the child must answer "yes, I can be claimed as a dependent by someone else" if they can be claimed by either parent, even if the parents don't want to claim them because they think it will result in better financial aid.
