You need to have all necessary DNS records in your Cloudflare account. So whatever DNS records you have at Namecheap, you’d have to manually type in here. That would explain why you’d sometimes get the Host Not Found – Your lookups sometimes go to Cloudflare name servers, and sometimes Namecheap.
As for DNSSEC, leave it off until all of your DNS issues have been fixed.
Then you can delete the Namecheap name servers at GoDaddy. Namecheap can continue to host your files, and Cloudflare DNS will point to the Namecheap IP address, as it seems to in your original post screenshot.