What is the ruling on fulfilling a vow to slaughter a sheep upon moving into a new house, and is it permissible to distribute it as the meat of a sacrifice (udhiyah) is distributed?

If the wording is "I will slaughter a sheep," this is not considered a vow (nadhr), and you are not bound by anything, because a vow must involve wording that indicates a commitment to an act of worship (qurbah). Therefore, you are not obligated to slaughter it at all, and if you do slaughter it, you may eat from it, give it as charity, and present it as a gift.