"Instead of heating to 74 degrees thermostat setting, the heater will only kick in at about 62 degrees and never get the home above 67 degrees." Does this same behavior happen when it's 25 or above? When it's 25 or above does the call for heat happen at 74 (thermostat setting) or 62? If heating starts at 62 even though the thermostat is set for 74, then something's wrong with the thermostat.
When it's below 25 and the furnace doesn't fully heat the house, does the main blower continue to circulate air even though it's cold due to no combustion happening?
Is the furnace located in a place that will be colder when it's colder outside, like an unheated basement?
If you can watch the furnace run on a 20 degree day look and see if after the fire goes out there is an attempt to re-light, possibly after a minute or so. When the fire goes out, does the inducer motor stop also?
If the temperature is affecting the inducer assembly then the furnace should completely stop... no air circulating in the house, no re-light attempts. If the furnace is going off because of overheating, then the main blower should stay on and there should be a re-light attempt after the heat exchanger cools down... maybe a minute or 2. You need to observe carefully to know what may be the problem. If it's very cold and you directly jumper, TEMPORARILY and with POWER OFF, the R and W terminals (bypass the thermostat) and it runs well, then it's the thermostat.