Both our logins in Mail show the same email address -- should be different

We recently changed email providers.
Both our new emails end in "gmail.com." But we use different names/addresses/signatures. I am the main login to the computer and the wrong address shows up exclusively only when my wife sends out a message (though not consistently!). Mail handles both the addresses I myself use (iCloud.com as well as Gmail) with no problems. Howcum? How can I fix this? (FYI: There was no problem with our previous email addresses, before the change.)
And further to the topic of Mail: often, Mail locks up and I have to Force Quit to get back to work with Mail.
And there's another peculiar habit it has picked up. After I select some old emails, right click and ask the machine to Send to Trash . . . it sometimes says it can't. "Could not be moved to Trash." Doesn't tell me why, though.
This is not on a level with solving the Ebola crisis. But I sure would like to get it under control.
Thanks for any help you may have.

Don't change a thing. You both already have separate accounts. When you set up her Mail account be sure you are logged into her account at the time. Also, you have to configure an SMTP or Send Mail server. That will be the same in her account as in your account if you both use the same email provider.
When you each send an email from your respective accounts they may show that the email was sent by the wrong party because you have not configured who to show as the sender in Mail's preferences - click on the Composing icon in the toolbar. Each account should be made different so that her mail is sent by her while yours is sent by you.
Configuring Mail for your email account
Apple - Support - Mac 101 - Troubleshooting 101
Mac 101- Mail (OS X Mountain Lion)
Apple - Find Out How - Mac Basics
Mac Basics- Mail (OS X Mountain Lion)

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