Will I be sinful for not completing the twenty days of fasting that I vowed as expiation for committing a sin, knowing that I broke my fast due to studying and revision, and that I began observing the Fast of Dawud in the month of Sha'ban?

You must repent to Allah and resolve not to return to the sin. You are also required to offer expiation for breaking an oath, which is feeding ten poor persons, clothing them, or freeing a slave. If you cannot afford any of these, then you must fast for three days, as Allah Almighty says: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended of oaths. So its expiation is the feeding of ten poor persons from the average of that which you feed your [own] families or clothing them or the freeing of a slave. But whoever cannot find [or afford it] - then a fast of three days. That is the expiation for your oaths when you have sworn" (Al-Ma'idah: 89). As for a vow of obstinacy (nadhir al-lajaj), it is not obligatory to fulfill it; rather, its ruling is the same as that of an oath.