Starting date months after offer

So I got a job offer back in December 2018, accepted it, and the starting date is in June. Since then I've done a background check, official transcript, got a welcome package and that's it. Would it be strange of me to check in that everything's still good to go or if there's anything else I need to do beforehand? How would you word that in email? I'm getting antsy cause I want to tell my current supervisor that I'm putting in my notice soon.

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