Animation in the latest official release can be a bit tricky to deal with if you need multiple actions at once.
The Godot 3.1 dev. builds though (in which an official alpha should be coming soon) has vastly improved animation capabilities that should allow for this. They are already usable for game work if you can deal with having more errors than usual.
https://godotengine.org/article/godot-gets-new-animation-tree-state-machine
If you really need to stick to the stable build though, you can always try breaking parts of your model into different objects and animating them independently (Godot's scene system is very good in keeping such setups together).
