"how many real life FA choose to leave the Yankees or Red Sox?"
That's because the Yankees and Red Sox have all the money. In real life, there is almost no way that leaving the Yankees or Red Sox can mean anything other than making LESS money. The exception to this would of course be leaving one for the other. Which Johnny Damon did just a couple of years ago.
A better comparison is the NFL (or NBA, etc) because these are salary capped leagues, which (essentially) is what HBD is. And the answer to the question for the NFL is that basically everybody on the Super Bowl Champs who is a FA leaves, because the Champs' #2 WR will get #1 WR money on the FA market, the Champs' backup RB will get a starting job and money somewhere on the FA market, etc, and the salary cap prevents the Champs from matching all of these contracts. And they almost NEVER take less money to stay.