What did you do with your gmail history? Do you use a phone client for your email with webnames.ca?
I can only assume you mean any emails in the inbox or other folders. I have an email app on my home PC that downloads all locally. I can move those into other local folders on my home PC. I never keep anything on cloud servers. I also never used Gmail extensively. We have a local ISP and have used their email for years but knowing we were switching off that ISP (Bell based so getting more expensive every year), we had to tank our ISP emails. I wanted something like a Gmail, universal, not tied to an ISP, but that was more secure, not a known snooping company who's admitted to reading people's personal email. Having your own domain name was a bonus.
I'm assuming you mean what email app i might use. I actually don't use any email app on my phone except for Gmail. To login into my Webnames i use the browser directly. There are email apps one could use though. Find one you like and trust.