Referring to yourself and your husband

How would I say "My husband and I..."
Such as "My husband and I are living in Japan"
I feel like 私の夫と私 isn't quite right.

It depends on the context/flow of conversation, but if you say “otto,” that makes it pretty clear it is YOUR husband, so you can skip “watashi no.” Also somehow (as I am thinking how I would phrase it myself) I would probably say “I am living in Japan with my husband” instead of “my husband and i…”

Now there are different preferences about how you refer to your husband, but if you use “otto, 夫と二人で日本に住んでいます。夫と日本に住んでいます(I am living with my husband/two of us are living in Japan.) would be pretty natural. If you use “shujin,” 主人と日本に住んでいます(I am living in Japan with my husband). Once again “shujin” makes it clear it is your husband.

Another example
主人と私が知り合ったのは10年前です。(My husband and I got to know each other 10 years ago.)

「夫と私」/ otto to watashi would be the more natural way.

While otto translate literally as "husband," it's generally used for referring to one's own husband. When talking about someone else's husband, it's more natural to say "danna-san" or "go-shujin," which are more polite terms than "otto."