What's the best OpenAI model for the most human-sounding outputs?

It’s not so much the model, but more about how you prompt it - which, in my experience, takes trail and error. Prompting is an art.

Models change - they come and go.

I suggest you learn about prompt engineering. There are multiple OpenAI sources on the subject. Here is one:

developers.openai.com

Prompt engineering | OpenAI API

Learn strategies and tactics for better results using large language models in the OpenAI API.