I did some more testing and realized there were a few other problems. For one, request headers are not exposed by default in Apache variables, so you must use the SetEnvIf directive.
And finally, I changed the proxy to use sockets. When using a simple HTTP proxy, it would still use the old Cloudflare IP. I added the web.socketed template in addition to the cloudflare one, and rebuilt.
Final config:
SetEnvIf CF-Connecting-IP (.*) real_ip=$1
RequestHeader set X-Real-IP %{real_ip}e
ProxyRequests Off
ProxyPreserveHost On
ProxyPass / unix:/var/discourse/shared/standalone/nginx.http.sock|http://127.0.0.1/
ProxyPassReverse / unix:/var/discourse/shared/standalone/nginx.http.sock|http://127.0.0.1/
And now it works 
