Simple example of the Full Adder. Add additional Full Adders for extra bits. This is only one Full Adder.
Switches are, in order...
A - B - CarryIn
All AND Gates are 3 Input gates with 2 ANDs cascaded, forming 3 inputs. All first inputs are connected to the first (left) switch. All the second inputs are connected to the second (middle) switch. All third inputs are connected to the third (right) switch.
Also, all inputs are categorized by height. The inputs flush with the AND Gates are A inputs. All the B inputs are flush with respect to the bottom of the first input (or one 4x4 square) and its positive end. The same is for the third and is with respect to the second input (B). This is the CarryIn input.
Note that only one AND Gate does not have a NOT Gate.