You can continue using your standalone Microsoft Publisher for as long as it remains installed and activated on your current computer. Microsoft ending support does not automatically disable your paid version. The program should run normally. The risk increases over time, especially if you replace your computer or upgrade to a new version of Windows, because older programs eventually lose needed system components.
Even after support ends, your installed copy will still open and save files, although it will not receive fixes, improvements or security patches. As long as your current machine remains stable, there is no requirement to stop using Publisher immediately.
The biggest challenge ahead is long-term access to your existing documents. There is no official compensation or free replacement for users of perpetual licenses. Microsoft views your license as fulfilled because you purchased the program with no guarantee of future support. Because of this, the best strategy is to keep Publisher running while gradually planning for migration.
There is no perfect replacement, but you can refer to https://www.reddit.com/r/microsoft/comments/1lmszyn/alternative_to_publisher/ regarding some of the most commonly used alternatives.
You also do not need to convert your thousands of files immediately. My suggestion would be to convert on demand. When you open a Publisher file you need in the future, export it to a format that will remain readable everywhere. The two formats most people keep are PDF for finished documents and DOCX if they need to edit outside Publisher, though some formatting may shift.
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hth
Marcin