I did a test in my SharePoint Designer 2013, saving edit form works well in it.
Whether your account with MFA enabled?
Try use another account to save edit form to SharePoint Designer 2013, check whether the issue exists.
You could follow below tips to troubleshoot the issue.
1.Clear cache. In the local computer, remove all files found in each of the following folders. Then open SharePoint Designer 2013 and enter the account again to see if it works.
- %APPDATA%\Microsoft\Web Server Extensions\Cache
- %APPDATA%\Microsoft\SharePoint Designer\ProxyAssemblyCache
- %USERPROFILE%\AppData\Local\Microsoft\WebsiteCache
2.Verify that SharePoint Designer 2013 is updated with SharePoint Designer Service Pack 1 and the August 2, 2016, Update for SharePoint Designer 2013.
If the answer is helpful, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment".
Note: Please follow the steps in our documentation to enable e-mail notifications if you want to receive the related email notification for this thread.