I set up a domain with Namecheap
I set up a private email server also Namecheap on that domain.
I changed nameservers on Namecheap which I got from Cloudflare
I copied all DNS records off Private Email email service from Namecheap to Cloudflare
as mentioned on the page
https://www.namecheap.com/support/knowledgebase/article.aspx/9967/2176/how-to-set-up-dns-records-for-namecheap-email-service-with-cloudflare-cpanel-and-private-email/#pe
But it sends emails and does not receive emails.
Make sure your email-related DNS records are set to 
DNS Only.
After you confirmed what @sdayman said, if it is still not working, consider this
Delete any MX records that are pointing to eforward*.registrar-servers.com.. You may still have some old records from the built-in email forwarding of namecheap which aren’t remove so your email is sending to their email forwarding service, instead of their private mail service. These two are different.
Also, let us know your domain, it’s easier to support once we know your domain and can tell imediately if something is off the chart
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