I've worked with pieces like that before. Are you saying the seams were serged so its coming off in pieces, or that when you cut it apart it starting coming off like that?
Two very different problems.
I'm not sure. On one of them I cut the seam out of the sleeve and started unraveling and the pieces would just stop once I got to the other end. I'm guessing that one had serged seams. On the other one the yarn was knotted at either end of a rib, it's sort of hard for me to explain properly.
I've been trying to think of other uses for the sweaters if I can't use the yarn. One is cotton/linen and the other is silk/cotton, so I wouldn't be able to felt them.