Heads up that TalkTalk MailPlus is a £50 a year subscription for eligible TalkTalk Mail accounts and that subscription covers up to 5 email addresses / mailboxes so is suited to a family using TalkTalk Mail.
In one reply, I've just saved you £50 a year because you do not need two MailPlus subscriptions - but do read my NB heads up footer note below.
All that you need to do is verify via the subscription link (below) that one of the email addresses and its current email password is valid and available for a subscription. The second email address will need to be separately verified as regards its registered user and added to the new TalkTalk Mail Plus My Account by TalkTalk Community Support right here.
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Let us know if there's an issue using the Subscribe to TalkTalk MailPlus link as this should get the first address subscribed and you can then come back here to get the second one added to the MailPlus customer MyAccount so that it comes under a single subscription.
NB. Just a heads up that all email accounts (max 5) via a single MailPlus subscription / MailPlus MyAccount will be registered to and under the control of the named account holder. So if this is a problem or could be a problem if the named account holder was not available for any reason then maybe you would need to consider separate MailPlus subscriptions.
In relation to being asked to sign in repeatedly I would expect that is due to the password being considered weak. I recommend 12 - 15 multicase letters and numbers and at least one approved symbol.
Check via the TalkTalk Mail webmail sign in and the user avatar top right of TalkTalk Mail and its menu item 'Update your reset details' that you do have password reset options listed correctly and saved ready for a password upgrade.
The Reset password now button in the help article Changing your email password will take you to password reset and recovery.
So long as the email address is still active on the TalkTalk Mail platform and you have set up password recovery options, in advance, that you recognise from the *****hints, then you can send your alternate email address a link or your mobile a 6-digit code to authorise a password reset.
