Is my donation of money, which I did not specify as Zakat al-mal, and of which the receiving organization spent the surplus on its charitable projects, considered an acceptable Zakat? And what is incumbent upon me to do now?

Do not give zakat with the intention of donating; rather, it must be given with the intention of zakat, because it is an obligatory right. The amount donated to the aforementioned entity is not considered zakat, because zakat is only valid if intended at the time of its disbursement. It is not sufficient if it was disbursed without the intention of zakat and then intended afterwards. If you possess wealth on which zakat is due, you must disburse its zakat because what was given with the intention of volunteering is not sufficient to fulfill the obligatory zakat. If you disburse it with the intention of zakat, you must inform the person you authorize to spend it of this, to ensure it is spent in its legitimate channels.