Forgive the long-winded question, but some background is necessary.
I activated full zone setup at Cloudflare partner LunarPages for two of my domains, one of which is an addon domain to the other, if that matters. That worked fine, but I wanted to see the origin IPs in the cPanel error and visitor logs rather than Couldflare’s proxy IPs, and I want to be able to ban IPs in .htaccess. For whatever reason, LunarPages will not install mod_cloudflare.
LunarPages suggested I switch to CNAME setup. After doing that, I get Error 1016 on all pages. Cloudflare diagnostics tell me to "add a verification TXT record to your authoritative DNS," but neither Cloudflare nor LunarPages seems to give me access to change DNS records.
I just transferred all my domains to Cloudflare as the registrar, if that could be part of the problem. The transfers were completed this morning.
These domains are on the Free Website plan.