The closest word I can come up with is “dynamic”, but that has positive connotations. I’m looking for a word for someone whose voice is dynamic to the extreme–overly varied intonation.
The best I can come up with is pitchy, but that is more appropriate for singing.
There is also affected, but that’s overly broad–I think of a put on accent when I hear that as much as overly variable tone.
Maybe you are looking for singsong?
Merriam-Webster: “a way of speaking in which the sound of your voice rises and falls in a pattern”.
I would say it’s usually understood as an undesirable trait.
This video gives an (exaggerated) example.