You’ll probably hear both forms in spoken American English, but to my ear, the grammatically correct version would be: “I went to school, and then I went home.”

You are correct! You can use “then” as an independent conjunction word. The correct sentence is this : I went to school. Then, I went home. If you say, “I went to school, and then went home”, this is also correct. However, if you say, “I went to school. Then, went home.” This is incorrect because, “Then, went home” is not a complete sentence. You need to put I in front of went to have a subject.

Both sound good in conversation. But in an essay or something, I would say the second one