Should I schedule a call/meeting or just ping him and ask if he has a minute to talk? My issue with scheduling is that he'd reject the request and delay/defer it to a later date because he's so busy or whatever.
I do have a 1x1 with him on Friday but I was instructed by the new place, out of precaution, not to give my official notice until my background check and drug screen have passed.
You're over thinking again. You absolutely do not need to do anything other than give written notice. If you feel it's right to have a phone call about it, then go about that however you feel is best, but it's a pure courtesy, so there's no rule as to how you do it. As an employer, I always hated when people tried to talk to me first before giving their notice. If it's a done deal, just give it to me in writing and stop wasting my time with your platitudes and feelings. If I care about what you have to say, I'll schedule an exit interview, otherwise, don't make some big dog and pony show about it.
My point is not that your manager wouldn't appreciate a call, it's that maybe they would and maybe they wouldn't. You know your relationship with them best, so do what feels right to you because at the end of the day, it really isn't all that important. It's just business.
Don't let this take up real estate in your brain, you have enough going on in there already. Just establish the official notice date, and go from there.