Is the Aqiqah considered legally acceptable even if not all of its meat is distributed, and the family is compelled to eat part of it due to need? And is it obligatory to buy new meat, or is it sufficient to distribute what remains of it?

It is permissible for a Muslim to distribute the 'Aqiqah in any manner: in thirds, or halves, or by eating it all, or by giving it all in charity, or by making a feast with it. All of these are broad options. Some scholars have held that making it a feast is disliked. The best approach is for the owner of the 'Aqiqah to eat from it, give some in charity, and give some as a gift. The Sunnah of 'Aqiqah is fulfilled by the slaughtering.