Set to 68 degrees....it keeps heating to 78 degree, or more...what's up...?. I reset it to 68, and it turns off..?

Thermostats have multiple issues when you start checking them against thermometers.

1) the built-in thermometer may not match the set temperature because one or both need calibration. Compare the built-in to an external thermometer and see if there is any adjustment to make them match. Record the setting on the thermostat and the temperature when it starts and stops -- there should be a 1-3 degree range which may be adjustable. The set point should be in the middle of the on-off temperature spread.
2) the thermostat has a heat anticipator setting which may be set poorly -- turning off the heat before it reaches the set point - the fan normally runs for 1-2 minutes after the heat source is off. The anticipator is used to stop heating before reaching the set point as there are still a lot of BTUs coming out during the fan run-down -- usually 1-2 minutes.

Yes there is a one degree tolerance, in fact my heater just kicked on and it is set to 67 and the inside temperature reads 67. It is also a Honeywell.

Its probably not the thermostat. The air conditioner may be stuck on heating mode. If its a reverse cycle, the reversing valve may be jammed so even when the thermostat is calling for cold air, it stays on heat. If the machine is on heating mode, it should be blowing cold air out of the outdoor unit of the air con. If its on cooling mode, the outdoor should be blowing hot air.

If your thermostat has an emergency heat setting this means you have a heat pump system.
Yes you need to set the thermostat to heat for heating the house.
The emergency heat setting should never be used unless your outside unit is not functioning.
The emergency heat setting overides the heat pump (the outside unit will not be energized).
The emergency heat setting will run the auxillary heat (heat strips) only.
When the thermostat is set in the heat position the heat pump will run to heat your house.
If the heat pump can't keep up (extreme cold) then then the auxillary heat will kick in to augment the heat pump until it satisfies the thermostat settings. If your room temperature gets 2 degrees below your thermostat setpoint, your auxillary heat will kick on. Example, say you are going to be gone a couple of days and you turn your heat down to 60 degrees. When you return and it is 60 degrees in your house, you turn the thermostat up to 70 degrees. Since you are turning the temperature up 2 degrees or more (10 degrees in this case) from the 60 degree room temperature then the heat pump will come on and the auxillary heat strips. The unit will continue heating like this until the temperature in the house gets to 69 degrees and the heat strips will turn off and the heat pump will continue to run until the 70 degree setpoint is reached.
Hope this helps.
SeagullAC

Dear friend! Any thermostat, it's calibrating to cut off at the end of 2nd beginning and cut in 2nd ending of temp.
Ex: In heat mode, If you set @ 68 degree, it will cut off 70 degree beginning and cut in at the end of 66 degree.
Hope it's helpful to you?

You did not make it clear if the thermostat is a programmable unit or not. I am wondering if the thermostat is not set on the correct heat anticipation setting. A heat anticipater is a small heater that helps the thermostat turn off before it over shoots the desired temperature setting. If the thermostat is set for a 5 degree setting, it will turn off at 63 degrees if set at 68. If the anticipater was set at 1 or 2 degrees, it will shut off the heater at 66 or 67. Newer thermostats will only allow the thermostat to cycle 6 times per hour so if you never reach the desired temp, you have to wait 10 minutes before it will allow a call for heat. Check the installation instructions to see if there is a setting where you can get the anticipator temperature closer to your set temp. Good luck.

Thermostat shoud be on auto and cool. Setting is normally set at 75-78 degrees f.In the winter, keep it on auto and heat. Setting should be kept around 60-65 degrees f.

Nothing is wrong; the air circulation in the room affects the thermosistor slightly in your thermostat, so does shade/heat etc.

You can adjust the position of the room air sensor found on the back of the unit between the heater and the wall. It is a black rubber bulb with yellow wires. There is a few inches of extra wire you can extend higher to get a more accurate reading of room air. While you are back there, check for drafts that will give you false readings.

Air conditioners rely on a difference in pressure between the "high side" and "low side" in order to perform. Of course, a/c's are designed to work in the heat. Having a cooler outdoor temp of 70 degrees probably is not sufficient to maintain adequate head pressure and will result in reduced cooling capacity. Also, since it was 68 in the morning in the room, it is possible that the indoor coil began to freeze up. Basically, you have a cold coil that you keep warm by moving air over it. If the air doesn't have enough specific heat (what your thermometer measures) it can allow the coil to get colder than 32 degrees and the condensate from all of your humidity will begin to freeze on the coil. My recommendation would be to check the a/c on a day that is 80+ degrees. If you have an avarage indoor temp of 70 degrees you should be able to look for the air coming out to be approximately 20-25 degrees colder than the air being sucked in.