Hi cheics,
As a first step, please remove any Font Awesome 5 Free/Brand fonts from your system, and then reinstall the new font files (v 5.5.0) from the Desktop version:
https://fontawesome.com/how-to-use/on-the-desktop/setup/getting-started
After installing the OTF font files from the above link, open up your file, go to “Publish > Generate HTML Files > Fonts”, and then update the CSS link to reflect the new version:
https://use.fontawesome.com/releases/v5.5.0/css/all.css
Now, it looks like the newer versions of FA 5 have fixed the issue of the Solid icons not rendering in the output with Axure RP, so please remove the Font Awesome 5 entries you have in the Font Mappings tab.
Please also note that Axure RP does not work with the ligature method of adding Font Awesome 5 icons, i.e. typing out the widget name. You’ll see this working on the Axure RP canvas, but when published, the icons will not render properly and will instead show the name of the icon you are referencing. To resolve the issue, you’ll need to copy & paste the icons directly from the Font Awesome icon site into Axure RP:
https://fontawesome.com/icons?d=gallery
If this still isn’t working for you, please post your .rp file here so that I could investigate further. Thanks!