Hello and welcome,
When you're set up to "save" via GH, it does nothing to save to your local file. It's sort of like you have to pick one or the other. There are various other saving options for saving to your local system. AFAIK, none of them operate at the same time as the GH saver.
The simplest way to save a local copy is to go into your saver plugins (under settings, plugins, savers) and delete your user name or password from the GH saver. Now click on the red check icon and your browser will save a copy of your file to your browser's download directory. If you already have a copy of your file there, it will save your file as a numbered copy.
In terms of a nearly seamless use, you can set up github to serve up your tw file. Search for "github pages". Your file will then have a url like:
You an then set up the GH saver to push your saves to the equivalent location in your GH .io repository. Whatever device you're on, just browse to
https://myname.github.io/myfiles/stroll.html, refresh, pop in the GH password token, and you're off and running. Save your changes. Then when you go to other device, just browse the url again and there you have it. Whenever you need a local copy, just disconnect the GH saver, click on the check mark, and you'll have a local copy in your download directory.
If you make changes to your local copy, you can reconnect the GH saver, and push them back to GH.
HTH