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1. Is it OK to rephrase the quote above, "My baby goes with someone new"? Which one is more common?
No, that would change the meaning, N. In the original, it's as if the singer is pointing to his girlfriend, saying "there's my girlfriend leaving with someone new".
Your example means, "My baby has left me and she is dating/going with someone new".
"She was my baby
'Til he stepped in."
2. What does 'step in' mean? Is it a common usage here?
It means 'became involved with her'. Iiiiiiiit's in use. You'd have to check a corpus to see how widely used it is.
Thanks in advance.
caress v. n. (source: Cambridge)
to touch or kiss someone in a gentle and loving way:
Gently he caressed her cheek.
bye bye love 
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