I had similar issue, i had Bitbucket SourceTree that i used to use to manage my local Github and Bitbucket repositories, but after installing local Git i started getting Git always asking for my username and password on push.
I tried installing different versions (newer & older) of SourceTree but that didn't work at all, google for long but didn't work as well, i tried re-creating repositories using different methods (such as SSH) but didn't work as well, in fact it almost caused me my changes to be lost - so i completely understand your frustration.
So to keep things as short as possible, here are the solutions that worked for me (as i had that problem multliple times on multiple machines):
The best solution is to Uninstall your local Git and install it again (that fixed my problem somehow and i had 0 data loss with it and my repositories in both Git & SourceTree were fine)!
I found on the internet that using SSH instead of HTTPS as repository origin worked for some so you can try that out redirect your local repository to SSH something like this command should work:
git remote set-url origin email@example.com:username/repo.git
Hope this helps solving your issue.