JJ-RS

Yes, at that time there were no clear instructions available. Having checked in the evening before resulted being on the preliminary passenger manifest and that was enough for CPB to cancel the ESTA. Apart from that I can confirm that documentation will be checked during check-in at the airport (in my case it was in Frankfurt) by the airline's embassy / CPB liaison.  

Maybe worth mentioning as well: During a subsequent trip with the CR-1 visa in my passport they still had to check my marriage cert, the visa itself was not sufficient even though it stated that I was exempt from the proclamations...

I simply applied for a new one which was approved in 2-3 hours 🙃

agreed - I had a certified copy of the marriage cert and a standard photocopy of my wife's passport with me.

Some first hand experience from the weekend: The previous replies in this thread are correct, you'll mainly need to have your marriage certificate and a copy of your spouse's passport as proof. However, I personally ran into several issues, one of them being that my ESTA got cancelled (not revoked, but set to expired) roughly 12 hours before scheduled departure and as I noticed that just in the morning I didn't have time to get this corrected and ended up moving my flight by a day. Seems like CBP are proactively canceling your ESTA once you appear on a passenger manifest. 

At least with United, there's no way to check in online, so you'll end up doing it the old-fashioned way at the airport. They'll put a remark in their system that you're traveling as immediate family member after verifying your documentation. Don't expect this to be your typical 2 minute check-in process; in my case my passport was handed over to another person who then made a phone call and ultimately confirmed that I'm good to go.

Over in the US you'll have some health screening before actually going through immigration. Not sure if this was related to traveling under these circumstances or attributed to the fact that I traveled after NOA2 or something different, but I ended up in my first secondary interview in 5+ years. YMMV.