HIDE YOUR ONLINE STATUS
If you're getting unwanted messages through Facebook's Messenger chat system, there's an easy fix. Load your Facebook page, then take a look at the Messenger list on the right side. If you don't see it, click where it says Chat in the bottom right corner of the screen.
Click the gear at the bottom of the list, then click the "Advanced Settings" link. From there, you can turn off chat for everyone or choose which friends can and can't see when you're online. It's a low-effort way to politely keep people from bothering you.