How to solve this task that relates to method of superposition
What did I do wrong to get the incorrect solution?


Two problems. Problem #1 is a trivial thing -- in your third diagram you show current flowing into the positive terminal of your voltage source, but current should flow out of your source. (Your equation says that all the reactive elements cancel out, so if we're only worrying about the voltage source and the resistor we can see that \$I_A^{,,,}\$ should be flowing in a direction opposite of what you've drawn here.)

The second issue is that in your analysis here you get the same answer for \$I_A^{,}\$ and \$I_A^{,,}\$, but they are in fact different. It would be nice for you to deduce the answer on your own, so I'll use the handy "spoilers" feature so that you can have a chance to try it on your own first, if you'd like to.

Hint 1:


   If you redraw your first two diagrams, but show the complete path taken by the current, you will see that these two diagrams have different current divider circuits at work.


Hint 2 (more or less a full spoiler):


   First diagram: $$I_A^{,}=I∗\frac{R−jX_c+jX_l}{R−jX_c+jX_l−jX_c}=1+i$$ Second Diagram: $$I_A^{,,}=I∗\frac{R−jX_c}{R−jX_c+jX_l−jX_c}=1-i $$


Combine that with the result of fixing problem #1 above, and you obtain the correct answer.