Trying to download High Sierra 10.13.6, the system says the file is damaged and can

Naw, it's just that you have an old copy already in Applications, there are 2 cures...

Disconnect from the Internet & set the Date back to 2015, then you can run the Install MacsOS High Sierra App in Applications.

Or... trash the App in Applications then a new copy can be downloaded.

When you go to the High Sierra page on the app store does it say Get or Open.

If it says Open then the App Store is detecting a copy of High Sierra on your mac.

Look in your Applications folder for Install macOS High Sierra.app, move it to the

Trash, empty the Trash.

If you have a suitable cable, either USB or Thunderbolt to Ethernet plug it into your

router. Restart your mac go back to the High Sierra page in the App Store

and try downloading High Sierra.