I keep trying to change my email address but it will not update and the old one keeps appearing

You have to have the email address changed by the security team.  If someone could just change an email address, it would make things too insecure.  The team checks to be sure it is you making the change and not someone trying to get into your information.

Submit an account recovery Request with proof of your identify.  It will usually be processed in 24 hours.

https://ttlc.intuit.com/turbotax-support/en-us/help-article/security-risk/account-recovery-request-turbotax-account/L3GhqZLVE_US_en_US

Are you trying this way?  How to change your email
https://ttlc.intuit.com/community/account-updates/help/how-do-i-change-my-turbotax-account-email/01/26385

This is so frustrating!  I've been trying for two weeks to change my email address because I want to close a legacy email account, I don't want to change anything else.  All the answers (here and on other sites) say to use the edit function in the security tab, but it doesn't work, at least for me.  I called the help desk and was told try it without my VPN on (I was even asked why I was using a VPN!), or that I should just use Chrome because it always works. Well I tried it with and without the Nord and Norton VPNs, with Firefox, Edge, and Chrome, and with the popup blockers disabled; nothing works!  All I want to do is get my emails sent to another address, but the edit function doesn't work on my computer.  All I get is the 'we need to verify it's you, please refresh' message or "this page isn't working" message.  I have no desire to entrust a photo ID or other government issued document to a private company that does not need it for such a simple change.  If TurboTax doesn't allow its users to self-change their email address, just say so and I will move on!

Contact TurboTax support and speak directly with a TurboTax support agent concerning this situation.

See this TurboTax support FAQ for a contact link and hours of operation -https://ttlc.intuit.com/questions/1899263-what-is-the-turbotax-phone-number

I contacted support earlier, they were not of much help.  They advised that I not use a VPN or that I should only use either Chrome or Safari; or that I should file an account recovery form - even though I still have access to my account.

It did occur to me that my Norton AV might be preventing something, so I signed in with an iPad using Safari - same result.

At this point, I am resigned to either set up another separate account with my new email address, or try another software package.