Game is all about controls, which are few and presented in help text to the left. I would feel bad giving more instructions about that :)
Why would axes push you up when you remove the axe from the ice? They push you up when you pull the axe down, which is only logical.
About spamming both the buttons: I tried to nerf this behavior with the stamina system, but I still feel the same way: there is no big reason not to use both hands. Which is probably fine, but leaves the core gameplay way too certain in its rotation for the player :) I am still thinking with the back of my head about other options, but still have not come with something easy yet worthy.
Thanks for sharing, and thank you for the warm words :)