Does angle of attack change in a sustained climb?
We have seen that 


  A climbing aircraft needs less aerodynamic lift than in horizontal flight, not more.


this means that if we keep constant airspeed and if we do not alter the configuration, the angle of attack will be (slightly) smaller, at least until the air density does not change.

As your assumption is that


  You apply full power


to enter the sustained climb, you will have even more thrust during the climb, and this will reduce even more the needed lift. This, together with the fact that you are slightly accelerating, reduces even more the angle of attack.