Is there a method for finding the net-ionic equation?
But apparently the correct molecular equation is $\ce{6KCN\+AlBr3 \to 3KBr\+Al(CN)6}$


That's not correct, what happenend to the other 3 potassium atoms?

It could form $\ce{Al2(CN)6}$ but you can't exclude $\ce{Al(CN)3}$ without further information beyond balancing of equations.

There are two main steps to balancing equation:


Balancing of each type of atom
Balancing number of electrons


If you've done both of those correctly, you've done all you can with balancing alone. 


  $\ce{Cl2\+2NaOH \to 2NaCl\+Cl2O\+H2}$


Above is clearly wrong because number of oxygen atoms doesn't balance


  $\ce{Cl2\+2NaOH \to 2NaOCl\+NaCl\+H2O}$


Above is clearly wrong because number of oxygen atoms doesn't balance