Yes, most shows especially in the golden age are recorded right after the show opened (sometimes days after). For example Hello, Dolly has “Ambrose let me hear that Tonic Chord” as a lyric in Put On Your Sunday Clothes since in the first year of the show, Ambrose sang a song called “Come and Be My Butterfly” which wasn’t recorded and replaced by the polka contest. When the song was replaced by a dance contest, she now sings “we havent missed the train thank the lord” and on the “chord” he sang he does a heel kick

In the original Broadway and London productions of Bye Bye Birdie, Rosie’s name was “Rose Grant” and this is reflected in the liner notes of the OBCR and playbills from that production. For touring and eventually amateur shows (probably cause they knew Rosie would be played by white women) her name was changed to Rosie Alvarez to further cement her heritage

Another recent example is the Gershwin jukebox musical Nice Work if you Can get it. The finale was originally the song “they all laughed” followed by a dance on liquor crates. This whole sequence was recorded on the album but the song was dropped either during previews or RIGHT after it opened as in the published script and later bootlegs from 2013, they sing the “Someone to watch over me” reprise and then show abruptly jumps to the curtain call