My house was at 60 degrees when I woke up (it was 10 degrees outside). I checked the thermostat and it was set to 69. I heard it start up but when it turned off it had only raised the temp to 62. I turned the thermostat up to 74 and pressed hold. It kicked on again and stayed on quite a bit longer. It still shut off when the temp was only 65. It then stayed off for a good 15 minutes. I pulled the thermostat display off the wall plate, blew it out and put it back on. The furnace immediately turned on and heated the house to 67. I removed the wall plate again and replaced it. Once again it turned on. Does this sound like a problem with the thermostat or is there something else I should check?

You would need to see if voltage is switched with the temperature selected.
Red may be your hot, green may be the blower and white may be your heat wire.
If you placed a meter across the red and green at the thermostat. When the blower is operating, you should have zero volts across the red and green, because the connection is closed, you will have voltage across the red and green when blower is NOT on. Test the heat the same way. If the switching is done at a different setting, thermostat is bad.

Measure across the red and white, to see when the heat is called for.

I would start by cleaning the flame sensor in the furnace, and also checking/cleaning the drain system of the furnace.