The sitcom Roc, starring Charles Dutton, had an entirely different theme song in later episodes. The original theme was a version of “God Bless the Child” sung by the Persuasions (at least the lead singer Jerry Lawson) and the later theme was a totally different song by En Vogue.
And early episodes of Remington Steele had the Henry Mancini theme with voiceover by Stephanie Zimbalist saying things like “I still don’t know his real name.” Later episodes just played the theme without the voiceover.
You are correct; originally Nell Carter sang it, but it was replaced with a more uptempo version performed by a group.
Technically, not every season was different - there’s a story about that.
In the next-to-last season, the first few episodes have the “Mother Hale” opening used in the last season. However, the people who did the real Mother Hale mural on which it was based wanted too much compensation for it, so Cosby switched back to the previous season’s Apollo Theater opening (and added a credit for Erika Alexander, who isn’t in it.) Apparently, so many people thought that the episodes were repeats from the previous season that they started having Cosby and Raven-Symone appear before the credits to say that it was a new episode.