What should I do if drowsiness overtakes me during the Fajr prayer, and I don't know what I said or did, without it being a deep sleep that invalidates ablution?

If you catch the prayer with the Imam and are aware of what he is doing, then your prayer is valid. The Imam bears for you whatever you forgot or were ignorant of regarding the recitation of Al-Fatiha or the tashahhud in the rak'ah or the tashahhud. The fundamental principle remains that you have performed what the Imam performs.