Your present thermostat likely has a setting for swing or an
anticipator, but cycling may be the furnace itself, you havnt had it
cleaned or serviced so do that first.

You say your bill doubled in price well sure your gas is up 43% and its
a cold December, and you never cleaned your furnace or had it serviced.
Look at your attic and measure it and figure 3.5R per inch. It depends
on where you live but R 35 is not optimal in the midwest R 50-60 is, I
have alot more than that.