Install lite.computer on the new machine and paste in the same license key. That is all it takes.
Your license is not tied to a machine and nothing counts your Macs, so the old one needs no attention.
Your pages live in the folder you chose, not inside the app. Move that folder, or keep it in iCloud Drive or Google Drive, and point the new install at it.
What stays behind is what the app remembered locally about a Showspace, such as which tabs were open.
Still stuck? Write to support@lite.computer. A person reads it.