Is it permissible to give the salary increase—which was vowed to be given as charity—to one's brother to pay off his debt and meet his needs?

Your vow is valid and must be fulfilled. You may give it to your brother because he has more right to it, as the Prophet, peace and blessings be upon him, said: "Charity given to a poor person is one reward, but charity given to a relative is two: one for charity and one for maintaining kinship ties." You may use it to pay off his debt or give it to him for him to dispose of as he wishes, unless you specified a particular entity or person in your vow, in which case you must fulfill it exactly as you specified.