Regenerate Thumbnails Plugin successfully updated my thumbnails but image src won’t change to new image size?

Regenerate Thumbnails won’t do this for you. It simply builds all new sizes from your upload archives. The next image insertions will have the correct image source.

For your past posts, you need to run a search and replace in the database.

Of course, you need to make a proper backup and test the search and the matches before proceeding to any modification.

That said, this plugin is adequate for the task:

Search and Replace

A simple search for find strings in your database and replace the string. You can search in ID, post-content, GUID, title, excerpt, meta-data, comments, comment-author, comment-e-mail, comment-url, tags/categories and categories-description.