furnace not getting house up to temp: suggestions?

When the weather here in the mid-Atlantic turned cold in the last
month, we began experiencing a problem in our house we didn't have
last year. The programmable thermostat will call for a temperature
increase, the furnace will run for a few-to-several minutes, the temp
in the house will increase by 1-2=B0 and then the heater will stop
without getting close to the target temp.

In the past the furnace has run for ~5 minutes and then shut down for
a cooling period while the exhaust fan still ran=97I'm assuming that's
to keep the burners from overheating=97and then fire back up to continue
to heat. But now the system just completely shuts down and doesn't
fire back up again for long periods of time, sometimes a couple hours.

Another symptom is that we have the thermostat set to 65=B0 at night and
to come up to 70=B0 at 6:45AM, but when we get up at 7AM the temperature
in the house has been 62=B0. Why would it be 3=B0 colder than the minimum
overnight temp and 8=B0 colder than what's called for?

What I need is advice on what the HVAC people should be looking for
when they come out.

System info:
15 year old Trane gas furnace (don't have the model # at hand)
Rite-Temp 6036 flush-mount programmable thermostat
We had an AprilAire whole-house humidifier installed by a friend (HVAC
guy) last year, and he blew the motherboard on the furnace installing
it. We've since had two HVAC people out to service the furnace and fix
the motherboard problems.
We didn't have this heating problem last year=97the system worked just
fine.
Nothing has changed on the system since last year.
I hard-reset the thermostat and reprogrammed all the cycles, and it
still has the problem.