Other than Donnie, who's a pre-dynasty Yankee who you're sad didn't hang around and win at least one ring?
For me I think about Mike Stanley a lot and how sad it is that the Yankees let him walk in 1995, after he was one of my favorite players and quietly one of the most important factors in turning the team around in the early/mid 90s.
Then they traded for him again in late 1997, the only year for the rest of the decade when they wouldn't win the series, but he signed with the Blue Jays after the season. Heartbreaking bad luck for him.