You can export an entire note to yourself using the email icon on the share sheet.
1. Open a note
2. Tap the share sheet icon at the bottom left of the note screen
3. Tap the Mail icon
4. Fill in your email address
5. Tap the "Send" button at the top right of the email screen
This will preserve the entire note in an email that will remain available in the email app until it is deleted from your email items. This is the only way I've found to preserve a note before removing it from the Notes app.