Always show language status
Solution 1:
This can be done by enabling the keyboard shortcut.
Now in full screen mode there will be a pop up when you use the keyboard shortcut (cmd+space) letting you see or change the input language.
The advantage of this method is you do not have something floating on you full screen all the time, but only when you call for it.
Solution 2:
You can use ShowyEdge to show indicators for current input langauges.
https://showyedge.pqrs.org/
It also works in full screen mode.