Is delaying the Fajr prayer considered intentional if I woke up a little before Fajr and intended to sleep for a short period to prepare for the prayer after the excuse of incontinence was gone, then sleep overcame me and I woke up after the time of sunrise had begun?

If the time expires while you are reciting Al-Fatihah, you have caught the dawn prayer before its time expired, according to many scholars. However, according to the view that the time is not caught with less than a complete rak'ah, then you have missed performing the prayer within its time, and a person knows best whether the delay was intentional or not.