In order for us to better investigate this issue you need to add a little more non-personal information.

Please add the troubleshooting information to your post To find the Troubleshooting information:

• Open Help (or click on three-line-icon and select Help)
• Choose Troubleshooting Information
• Use the button Copy to clipboard to select all. Do not check box "Include account names"!
• Paste this in your post.

Application Basics

   Name: Thunderbird
   Version: 52.6.0
   User Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.13; rv:52.0) Gecko/20100101 Thunderbird/52.6.0
   Profile Folder: Show in Finder
   Application Build ID: 20180123185941
   Enabled Plugins: about:plugins
   Build Configuration: about:buildconfig
   Memory Use: about:memory
   Profiles: about:profiles

 Mail and News Accounts
   account1:
     INCOMING: account1, , (imap) imap.cox.net:993, SSL, passwordCleartext
     OUTGOING: , smtp.cox.net:465, SSL, passwordCleartext, true

   account2:
     INCOMING: account2, , (none) Local Folders, plain, passwordCleartext

 Crash Reports
   https://crash-stats.mozilla.com/report/index/bp-843214a2-65a2-4ccc-84c6-ed50c2170118 (1/18/17)

 Extensions
   Lightning, 5.4.6, true, {e2fda1a4-762b-4020-b5ad-a41df1933103}

 Important Modified Preferences

   Name: Value

     browser.cache.disk.capacity: 358400
     browser.cache.disk.filesystem_reported: 1
     browser.cache.disk.smart_size_cached_value: 358400
     browser.cache.disk.smart_size.first_run: false
     browser.cache.disk.smart_size.use_old_max: false
     extensions.lastAppVersion: 52.6.0
     font.internaluseonly.changed: false
     mail.openMessageBehavior.version: 1
     mail.spotlight.firstRunDone: true
     mailnews.database.global.datastore.id: f4bb976b-8794-3f48-8bd2-205b76878ae
     mailnews.database.global.views.conversation.columns: {"threadCol":{"visible":true,"ordinal":"1"},"flaggedCol":{"visible":true,"ordinal":"3"},"attachmentCol":{"visible":false…
     media.gmp.storage.version.observed: 1
     network.cookie.prefsMigrated: true
     network.predictor.cleaned-up: true
     places.database.lastMaintenance: 1520275096
     places.history.expiration.transient_current_max_pages: 122334
     plugin.importedState: true
     security.sandbox.content.tempDirSuffix: {82443c48-2d45-6742-a62b-9730ddacde22}

 Graphics

     GPU #1
     Vendor ID: 0x8086
     Device ID: 0x1927

     Features
     WebGL Renderer: Intel Inc. -- Intel(R) Iris(TM) Graphics 550 -- 2.1 INTEL-10.30.14

     AzureCanvasBackend: skia
     AzureCanvasAccelerated: 0
     AzureFallbackCanvasBackend: none
     AzureContentBackend: skia
     TileHeight: 1024
     TileWidth: 1024

 JavaScript

 Incremental GC: 1

 Accessibility

   Activated: 0
   Prevent Accessibility: 0

 Library Versions

     Expected minimum version
     Version in use

     NSPR
     4.13.1
     4.13.1

     NSS
     3.28.6
     3.28.6

     NSS Util
     3.28.6
     3.28.6

     NSS SSL
     3.28.6
     3.28.6

     NSS S/MIME
     3.28.6
     3.28.6

Something like this happened before, and it turned out there was a 3rd-party password manager interfering with TB passwords. Disabling the manger resolved the problem.

I have LastPass on Firefox but disabling it didn't do anything with TB. Even after restarting it.

Select Help/Troubleshooting Information, click Open Folder to open the profile folder, then close TB. Passwords are stored in the two files: key3.db and logins.json. Check the Properties of these two files to make sure they are not Read Only. If you rename them to, say, key3.db.BAK and logins.json.BAK, then restart TB, new files will automatically be created; see if you can now enter passwords and have them saved in Saved Passwords.

So I followed those steps, and it did bring up a Mail Server Password window for me to save my password. But once I enter that new password it brings back the same Login to Server IMAP Failed window. So while the new password shows up in Show Passwords, I still can't access my new emails.

What do you have for the User name in the server settings? Cox says this should be the part of your address before the @. Using the full address could prompt the kind of error you're seeing, even if the password is correct.

I uploaded a screenshot of the Mail Server Password Required window. It shows that my username doesn't include the @cox.net part.

The place to confirm it's correct is under Tools/Account Settings, Server Settings/User Name.

If all your settings are correct, as they appear to be, it may help to change the password on the Cox site, and then enter the new one in TB (after you Remove the existing one in Preferences/Security).

http://forums.cox.com/forum_home/internet_forum/f/5/t/20278.aspx