The outer jacket is to just hold it together and have a place to install the insulation without it hanging in the breeze. An internal leak is likely, otherwise, the shell wouldn't be rusted. Probably time to replace.
If there isn't any water flow (leaks), the gas usage wouldn't spike, as was indicated, usage might go up a little because of the added insulation of the mineral deposits, but it shouldn't spike. IF it went up, it would be gradual.
Last question, did you check actual gas usage, or just the price? The price could have changed radically, and you're looking at dollars verses volume of gas used.
While you are not using any gas appliances (stove, heat, WH, dryer, etc.) is the gas meter dial moving? Unless you have a standing pilot (and you can turn those off), the meter reading should not move, just like the water meter shouldn't. If everything is off, and the meter's moving...call the gas company! Quick!, and better do it from outside, not inside.