When your transom assembly is off the boat and you grab the steering arm should you be able to move the gimble side to side like you are steering the boat? Or should it only be able to move when your steering is hooked up and you turn the steering wheel?

should move by hand with no effort.

I can move it up and down like it is being trimmed, but can't move it side to side. What would cause this?

Not sure, possibly the top nut on the steering arm has been tightened to much pulling the gimble ring tight against the housing at the bottom pivot. There should be a .002-.010 gap at the lower pin location. Or the large top shaft pivot pin is seizng up in the upper bushing, but usually that will wear loose. Check for the gap at the bottom.