Indeed, we find many great tzaddikim throughout the generations who, despite their busy daily schedules, and being occupied with many communal affairs, still managed to reach great Heights, achieving lofty levels in their Avodas Hashem.
One such Gadol was the Chasam Sofer. The Chasam Sofer writes several times in his Teshuvos that he is so laden by his many duties that he doesn't even have a few seconds to swallow his saliva. Despite that, he still managed to deliver a daily shiur to his students in Yeshivah, write responsum to the many halachic queries sent to him as well as leading his community, tending to all their needs.
Similarly, the Netziv, in addition to heading the legendary Volozhin Yeshiva with immense devotion, also served as the Rav of the city. This obviously came with countless responsibilities. His daily routine included delivering several shiurim, both to his talmidim in Yeshiva and to the townspeople. Amazingly he also managed to write many phenomenal sefarim, covering a wide range of areas. Indeed, he signed off his Teshuvos "העמוס בעבודה" - the one who is loaded with work.
Similarly, Rav Shlomo Kluger of Brod ztz"l, was the Maggid in his city, entrusted with the task of addressing his fellow brothers with mussar and inspiration whenever needed. Yet, he also dedicated of his precious time to write a very large number of sefarim which cover all areas and aspects of Torah; Halacha, Drush, Pilpul, etc. Legend has it, that he authored three hundred and seventy five sefarim - the numerical value (gematria) of his Hebrew name שלמה.
Many other tzaddikim followed similar patterns. Still, they all seemed to find time for things which, naturally, seemed impossible. But they merited special Siyatta d'Shmaya which enabled them to fit all their activities into such limited time, as explained earlier.