When I add up the mass of 6 protons and 6 neutrons in amu, I get a mass that is greater than the mass of carbon. I thought that it should be the other way around, because I have not including binding energy when I add up the mass of the protons and neutrons.
Proton: 1.007276466812 u
Neutron: 1.00866491600 u
6(1.007276466812)+6(1.00866491600)=12.099
Carbon: 12 u
Why is this so?
You are getting the right thing. This is the binding energy formula.
$$E_{\text{binding}} = (M_{\text{constituents}}-M_{\text{BoundState}})c^2$$
When the constituents come together to form a bound state the total mass is lowered not raised. Binding energy is the energy corresponding to the mass lost by the constituents as a result of them entering the bound state.
