The error message that the installer is damaged is actually the App Store detecting that there is a copy of the Install macOS Mojave.app in your Applications folder. When the App Store opens at the Mojave download page I bet it says Open.
Look in your Applications folder locate Install macOS Mojave.app, move it to the Trash, empty the Trash.
Connect your mac to your router via cable. Restart your mac, Open your Safari browser and click on this link.
How to upgrade to macOS Mojave - Apple Support
Go to Section 4 and click on Get macOS Mojave,
you will now be redirected to the App Store, this time it should say Get or Download.
Do not use the mac while it is downloading or any other devices that share the same internet connection.