[Plugin: W3 Total Cache] I can't change the permissions of my files to 777

You should call your host to make sure you understand how to change file permissions, once you satisfy the notifications in W3TC most users find that everything happens automatically in terms of server tuning.

I have root access to my server. I connected via FTP and used CHMOD to try and change permissions to 777 (I normally have no issues changing permissions or on this server)

I can’t even delete the plugin from my server…

I understand file permissions. It’s very simple to use CHMOD to change the settings.

Can you please advise or no?

from my experience you are on shared servers of Hostgator,who normally do not allow 777 to file directories.better try to talk with them live chat and tell them to change them W3TC directory to 777.
source https://www.tospider.com

No, as I mentioned earlier I have a dedicated server and have root access etc.

I could change the permissions before, but after upgrade or somehow sometime along the line (because I didn’t notice the plugin wasn’t working) it stopped working and didn’t allow me to change permission files or even delete the files.

The plugin itself doesn’t do that. When you upgrade plugins via WordPress admin, the files are written using the user and group that PHP runs under on your server. As root you should be able to modify those permissions as needed, but you shouldn’t need to. I recommend deactivating the plugin and following the installation instructions over from the start.