As a practical matter, if the name is readily identifiable as a famous baseball player and your songs are about baseball, one can assume that the player's attorneys will find a relevant statute that applies to your situation to make it difficult and expensive for you regardless of whether they have legal protection. A primary question in trademark law is whether a name has secondary meaning attached to it. But as others pointed out, there may be other legal rights that the player may have, including rights of publicity.
Using a name like "Babe Ruth" or some other readily identifiable or relatively unusual name will likely attract an expensive situation for you. If you write a song about how much you loved going to the ballpark and mentioned a name of a favorite player, that is different. But it seems that you explicitly want to attract attention and promote your product / music through the use of the player's name. Such a strategy probably doesn't bode well for you.
Going forward first and then offering some money later is probably not going to work out well. Famous / accomplished personalities care greatly about protecting their brands. Look at Drew Brees and Tim Tebow who have an explicit association with Christianity. Brand owners certainly want to know what is being done with their brand and by whom and certainly before any use enters the marketplace. Throwing the famous player or celebrity some additional income if your song turns out to be a hit probably will not be flattering.
