Many gas furnaces can operate in low or high fire modes. If you unit is firing only in low, it's probably not producing enough heat to maintain temperature in your home. This would most likely be a thermostat problem because the thermostat should engage the high-fire rate once the temperature drops too far below the set point. WHen the furnace switches to high-fire mode, it should also speed up the draft fan. That is the fan that I am assuming you replaced that you described as "a small fan near the burners". It basically sucks the burnt exhaust out the heat exchangers and blows them outside (and also sucks fresh air through the heat exchangers in at the burners where it mixes with the gas to be burned) I am thinking that maybe a single speed blower was installed in the furnace, and you have a 2-speed furnace, so without the extra speed on the draft fan, the burner is not going into high-fire mode to make up the difference on the really cold night.
Are you actually getting warm air out the furnace? After it's been running for a while, put your hand at a duct and feel if you've actually got warm air coming in. A likely scenario is that a piece of ductwork may have come loose and is leaking. If the supply duct came loose, some or all of your warm air is blowing out into the unconditioned attic or crawl space. If the return air duct has come loose, chances are, the air coming out will not be as warm as it should (because the furnace is sucking cold air from outside, instead of inside) and the furnace is actually pressurizing your house, and forcing the conditioned air out at all the cracks and crevices. You should be able to tell this if you open an outside door slightly and you feel a wind blowing out the house.