Hi Michael! Yes. 해어지다 means to wear out as in "my grandma's sweater was all worn out". It can be "우리 할머니의 스웨터가 다 해어졌어요".
The one you heard for saying goodby is actually not 해어지다 but 헤어지다. They sound the same but are spelled differently. So 헤어지다 means to break up for lovers, or to leave or part for just normal relationships. Like 저는 그와 헤어졌어요 = I broke up with him, and 저는 어젯밤 9시 쯤에 친구와 헤어졌어요 = I left my friend at about 9 pm last night.
Hope it helps you!