Getting this issue where the Die State is not properly transitioning to the Spawn state no matter what changes I give it.
Is the death animation set to loop maybe, such that the "animation_finished" signal never triggers?
Have you tried placing a breakpoint in line 4 of the Die state's script (press F9 with the light highlighted in the script editor)? If the game execution never stops there, it means the line of code is never reached by the compiler.
The debugger's generally your friend in those cases. Here's more info about how to use it: https://www.gdquest.com/tutorial/godot/gdscript/debugging/