Why doesn't natural gas combustion release only one-sixth as much carbon dioxide as coal burning?
When you do these calculations be sure to keep track of your units and the actual reactions you are considering. 

Burning methane is not the same as burning the same amount of carbon plus twice the amount of dihydrogen. You need to check the specific heats of combustion and you need to make sure you keep track of the units used to measure the heat produced in the reaction (per mol, per kg or per mol of carbon dioxide?).

The heat of combustion (assuming all products are at STP) for carbon is 393 kJ/mol and for methane it is 889 kJ/mol. Both reactions produce the same amount of carbon dioxide if combustion is complete. So burning methane produces about 2.3 times as much heat per mole of carbon dioxide as burning pure carbon (anthracite coal will be close to this).