Google Takeout will export all your mail in mbox format which is easily handled by mail programs like Apple Mail and Thunderbird and should let you sync everything to a new service (if that service won't let you transfer it themselves). I'm using iCloud+ from Apple. It's cheap at $1/mo for 50GB of space and lets me have my custom domains. $3/mo will get you family sharing and 200GB of space (if you want multiple acco…
Is iCloud+ IMAP/Push?
Unlike gmail they appear to implement the full imap protocol