You should have installed two apps, the old version, where you can find your current notes and the new one, where you will need to copy them.
There are several easy ways to copy your documents to the new version:
1) If you use iCloud, open your files directly from iCloud Drive in your new Notes Writer
2) You can use “Files” iOS app to copy folders and multiple files from the “NotesWriter” folder to the new “Notes Writer” folder (both of them are the iCloud Drive default app folders). You can use the "copy and paste” method directly in Files app. Once copied, open the Notes Writer, go to the “On iCloud” location and you will have access to all the copied folders and files.
3) You can drag&drop your files using the iOS 11/12 drag&drop gestures to copy documents from the free app to the new one
4) If you have only a few documents, a quick way is to copy&paste the content of your documents from old version into new blank documents in the new app using the “Paste” option. (If you have images don’t use this option).
5) Move your documents using the “Send to app option” in the share menu. Open your free app, tap the “Select” button and select a document. Tap the share menu button on the top toolbar then “Send to App” option. Choose the new "Notes Writer” icon to send your document to the app
6) You can also use the iTunes File Sharing: You can connect your iPad to your computer via iTunes and use the iTunes "File Sharing" feature to copy your documents from free version to your local hard disk then from there to the new Notes Writer. There is an Apple page about the iTunes File Sharing and how it works: