Every widget that you connect with your Facebook profile receives the same access token from Facebook. When you disconnecting your profile from the Facebook app (by clicking "Disconnect" button in the widget) - the token gets revoked, and every widget that was connected using your Facebook profile lose connection to Facebook. So after that you need to go and connect all other widgets again.
If you want to delete some widget - just delete it without disconnecting it.
Please, don't forget to click "Save" button after connecting the widget.
Best Solution:
Go to the Facebook Feed Pro component settings and set "Facebook Connection mode" = "Token Manager". In this case the "Connect to Facebook" button in the widget will be replaced with "Token Manager" button. When you click it, the popup window will appear, showing your saved tokens. You can store as many Facebook Page's access token as you need and use them in any Facebook Feed Pro widget on any website. The Token Manager stores the tokens under your AllForJoomla.com account. When you use Token Manager to connect widget to Facebook, you won't need to update the connection on every website you have used it. It will be updated automatically when you will update it in your Token Manager. How the Token Manager works you can read here: https://allforjoomla.com/help/facebook-feed-pro/2379-token-manager.
Go to the Facebook Feed Pro component settings and set "Facebook Connection mode" = "Token Manager". In this case the "Connect to Facebook" button in the widget will be replaced with "Token Manager" button. When you click it, the popup window will appear, showing your saved tokens. You can store as many Facebook Page's access token as you need and use them in any Facebook Feed Pro widget on any website. The Token Manager stores the tokens under your AllForJoomla.com account. When you use Token Manager to connect widget to Facebook, you won't need to update the connection on every website you have used it. It will be updated automatically when you will update it in your Token Manager. How the Token Manager works you can read here: https://allforjoomla.com/help/facebook-feed-pro/2379-token-manager.