STILL don't understand full-adders? Help?

I'm trying to figure out how to create a basic calculator, using only logic gates as the brain. I've tried online resources, and I even bought a textbook(Introduction to Logic Design) on the matter. Neither helped. Here is an example of what I want to know:

A full-adder has 3 inputs: A,B, and Cin(carry from previous adder). And it has 2 outputs: S(sum) and Cout(carry out from the adder, in to the next) Let's say I wanted to add 2+2. In binary, 2 is 10. Two 2-bit numbers would mean I have 4 inputs, and need 2 full-adders connected(Cout of the first connected to Cin of the second) right? So if I input 1 into A, and 0 into B of both of the adders, from what I understand, this should add 2, and 2(10, and 10 in binary) thus resulting in 100(which is 4 in decimal format). Am I understanding it correctly?