Both N.Y. teams have a "Turning Point of the Game" contest with the "clue" given out during the "game highlights" portion of both teams post-game shows (Mets on WFAN; Yanks on WCBS-AM) & for the Yanks, the "clue" given out again on one of the WCBS-AM sports reports (on the morning after each game). The 2 prizes are awarded in 2 separate contest segments (to the first caller in each contest that correctly identifies each game's turning point) on WFAN on the morning following the game, usually one team's contest during the 10am hour & the other team's contest during the 11am hour.
I'm not sure of the sponsor; I'll update as soon as I hear the contest again on Wednesday morning 5/26/10.
I'm not sure about the "Inside N.Y. Baseball" segment (I assume this would air during the Yankees pregame show on WCBS-AM)...I'll listen more carefully in the future to verify.