Were Edgar Allan Poe and his adopted parents close?

Poe did not have "adopted parents". He had foster parents. John and Frances Allan took Poe in to their home and raised him but they never legally adopted him.

As to their relationship, Poe and Frances Allan and a loving relationship. She doted on him and he was devoted to her.

Poe and John Allan had a stormy relationship. He most likely never wanted to take Poe in to begin with. Then he refused to pay tuition for Poe to go to the University of Virginia or pay for any of Poe's gambling debts. They had a falling out and Poe left home and the University and went into the army. Later after Frances Allan died, Poe and John Allan reconciled for a short time. Once again Poe's lifestyle became a problem and Poe and John Allan had a falling out that was never repaired. When John Allan died, Poe was not even mentioned in his will.