Does the worshiper receive full reward if he prays Fajr after drowsiness has left him, knowing that he did not intend to get up to perform it at the time of the Adhan?

A Muslim should hasten to obey Allah and avoid obeying Satan and the soul that incites evil. The Prophet, peace be upon him, used to hasten to acts of obedience, and Allah Almighty commanded this, saying: "So hasten to all that is good," and "And hasten to forgiveness from your Lord." Laziness in prayer and preferring sleep over it are not characteristics of perfect believers; rather, they are characteristics of hypocrites, especially regarding the 'Isha and Fajr prayers. The prayer you performed fulfills the obligation, but the reward will be more complete if you rise for prayer promptly, striving against yourself to make it obey Allah.