Is it permissible to eat from the meat of a vow designated for the poor and the needy, and is it sufficient to give the monetary value of what was eaten instead of slaughtering another sheep?

A vow (nathr) is not binding unless it is articulated verbally. As long as you have fulfilled your vow by slaughtering a sheep and distributing it to the poor and needy, you are not required to slaughter another sheep. However, you must compensate for what you ate from it and what you gifted from it, because it was a vow specifically dedicated to the poor and needy (a compensatory vow), and it is not permissible to eat from it. The compensation should be given to the poor and needy, either by paying its monetary value, or its equivalent in meat, or by purchasing a portion of an animal in which you share in its slaughter.