This is what I've done: sync your Gmail account with a regular mail client and backup the mail clients database so as not to lose any old emails.

Step two create new email address with dedicated email provider, one known to have good customer support.

Step three, stop giving out your Gmail address, and over time update your email address on all the services you use.

You can't take your @gmail.com address with you to a different provider, I think that would break the way the Internet works.