The article mentions "full faith and credit" which is the US Constitution's clause that says each State must respect the laws of the other. But that has not been consistently applied in the past.
"Full faith & credit" was ignored by the US Supreme Court for decades in the case of interracial marriages. They were legal in some states, but illegal in the South. The Court refused to force Southern states to recognize interracial marriages from outside the South.
Likewise, the US Supreme Court has never forced states to recognize gay marriages from those that permit them. The Defense of Marriage Act (DOMA), while not superior to the Constitution, nevertheless gives a "sense of Congress" which the present Court is unlikely to overrule, and upon which it will continue to base its inconsistent application of full faith and credit.
I therefore see an appeal of this decision by the state of Louisiana. And if it goes to the US Supreme Court, with the current mix of Justices, Louisiana will win, and the gay adoptive fathers will lose. Thank you, conservative Republicans.