Should one begin by fasting to expiate multiple oaths if no poor people are available to feed, knowing that this might conflict with fasting the six days of Shawwal?

It is permissible to fast the six days of Shawwal before atoning for oaths, especially if the expiation is not by fasting. However, it is not valid to atone by fasting if feeding or clothing is possible, even if searching for the needy takes time. One must search for the needy or seek help from those who can guide to them, or delegate individuals or charitable organizations. Delaying the expiation due to searching for them is not blameworthy, and it does not necessitate resorting to fasting.