I have my domain set up to proxy the A record to my webserver. When a request for my domain hits nginx, it appears that the domain name is stripped from the record, and thus it doesn’t hit the correct nginx server block. When I do this with a domain pointing directly to me without proxying, I get the domain and it works fine.
DNS records
At the Cloudflare dashboard, or? 
Yes, even the sub-domain such as www might be shown only, and not www.example.com, if your question is related to the DNS tab of the Cloudflare Dashboard UI?
Using multiple domains with each separately using it’s own .vhost config file, or?
Wonder what you get in the log files on your server then?
That seems unlikely. Do you have a log snippet which demonstrates this?
Well, I DID have a log of it earlier, and now I cannot find it, and can’t reproduce the one for the domain that’s working actually having the server name in the log, so I’m going to have to close this and do more research to see why nginx is ignoring the server block for this. Sorry, and thanks everyone.