What is the ruling of Islamic law on handing over the amount to the needy person and him promising to pay it back to the legatee, and is it obligatory for me to pay the money to the legator or to the needy person who did not fulfill his promise?

The agent is trustworthy in what he is entrusted with, and the implication of trustworthiness is to hand over the money to the person designated by the principal, and to no one else. As soon as you pay it to someone other than the person you were instructed to pay, you become liable for it. The principal is still obligated to pay the debt because he did not deliver it to the creditor.