Must I pray Fajr immediately upon waking up, or is it permissible to delay it slightly to relieve myself and shake off sleepiness?

If you wake up after the congregational prayer has passed, hasten to prepare for prayer by attending to your needs and performing ablution. Do not delay or be lazy, lest the time for prayer expires. Hastening to perform prayer is a form of racing towards good deeds, and praying on time is among the most beloved deeds to Allah.