Regarding the issue you are experiencing, when switching hosts it's important to update your DNS records in your Ezoic account to match those as they appear at your new host. This is because Ezoic needs an up to date copy of the DNS records in order to know where to go to fetch your content and serve it via our proxy.
In your case, you will need to update the full list of DNS records (A, CNAME records, etc.) in your Ezoic dashboard.
To do this, you should locate your DNS records in your Hostgator account. They may also be called "zone files" and can usually be found in the "zone editor." Once you have located your DNS records, you should edit the copy in your Ezoic dashboard to match the DNS in your Hostgator account. You can update these manually in your Ezoic dashboard under 'Settings' > 'DNS Settings' > 'Edit DNS Record,' or you can upload all the DNS records through a .csv file format.
By doing this, Ezoic will continue to function properly while you migrate your hosting to Hostgator.
In respect to the name servers, you will need to change these at Hostgator to point to Ezoic, you can switch them in Namecheap to point to Hostgator if that is how you are transferring the hosting, it is where the new hosting is at that counts.
Please let me know if you have any further questions or concerns.