Back when the Moody Blues was actually a blues band (or R&B band), they recorded ‘Go Now!,’ a cover they mostly changed by adding that exclamation point to the title. (Denny Laine, the later Wings guitarist, played lead for them during those times.) It was released in January of 1964.
Then, once they began to write their own songs (with the additions of they went into a completely different sound and style, which is what most people remember.
Edited to add: Fleetwood Mac is another obvious one. Their first hit was ‘Oh, Well,’ and although it was in the style of their prior two albums— a guitar rave up song based on their blues roots— it was very far from their prior cover versions of classic blues songs.