As Cab had mentioned, it could be a problem at the furnace, not the T'stat.
To eliminate the T'stat completely, turn the heat cool switch to "off" and at the furnace terminal board, jumper R to W and see if the same thing happens. It is only 24 VAC but be careful not to short while trying this.
Gas heat? Check your outlet temperature and see if it is high. This would indicate a lack of airflow due to closed registers/vents, restriction through the air filter, dirty fan wheel etc.

Lol, I see the fear of electricity post after I had posted my response.

Turn the furnace off while jumpering R-W and that should eliminate any possibility of getting zapped!

Yeah, our furnace is up in the attic. I'm not as spry as I used to be so I'm not about to go up there to mess with it.

I am with you there! I avoid the attic space in the house and haven't been up there for years.
Did you have the T'stat off the wall for the photo? You could also jump R to W at the thermostat.
Don't want to talk you into doing something you are not comfortable with.

Okay, turn the furnace off at the breaker. Tie the R and W wire together at the thermostat. Turn the furnace back on at the breaker and see what happens. If the furnace runs and the house warms, it's the thermostat. If it short cycles, limit. And no worries about zapping anyone if you follow my directions. Plus you're on the low voltage (24v) side when working at the thermostat. Just make sure you turn the breaker off, remove the red (R terminal) wire from it's terminal, Remove the white (W terminal) wire from its terminal, twist the two together and turn the furnace back on. You'll probably hear the fan cycle for about 30 seconds when you power it back on then it'll start going into a heating cycle.

If you're not comfortable messing with it I get it. But with the breaker off for the furnace you shouldn't have any worries and 24v just gives you a nice little tingle. If you'd feel better, I can PM you my phone number and talk you through the whole process. I even promise not to yell DON'T DO THAT while you're messing with the wires.

Based on that, you're cycling off of limit. The Heat on indicates you've got a call for heat. The fact that it remains on when the furnace turns off tells me you're cycling off of limit. I'd call a heating contractor (well, I wouldn't because I am or was one) and have them check the system out. Make sure you check your air filter or filters before you call someone to come look at the system. I've seen some attic systems have a primary in the wall or ceiling of a home but also have one at the furnace.

Edit: One more thing. Make sure all of your supply registers are opened and none are blocked off too.

OK, got a call from the local electrician/heating/a/c company we've used for electrical stuff. They are swamped with calls from people without heat so we slid down the totem pole on getting them here before Christmas, which is fine with me. We have heat and I'd be in a world of hurt if we didn't. They said it sounds like the heater is working as designed. It may just need a thorough cleaning, which it's never had in the 10 yrs we've had it since it was put in. We don't use it like most people do/would. It only gets run if it's really hot or cold. It's not a set it and forget it system as far as how we run it. I doubt it has 1 full years worth of run time on it. So they'll call us after Xmas and set up a 'thorough service cleaning' to make sure everything is OK.
Thanks again for all the help.

Yours may not have batteries. It's possible for a thermostat to get power from one of the wires coming from the HVAC unit. If that wire isn't there, then you must have batteries. If the wire is there, then I think you can have batteries anyway which might (can't remember for sure and may depend on thermostat) help with settings in the case of a power outage.

Understandable. I probably need some sort of event, because 9 times out of 10 that I do work on electrical, I don't even turn off the breaker. I've recently rewired lights, switches and outlets in 2 bathrooms with the breakers on.

yeah. "Ground" is not automatically ground.
It's often several circuits connected to the same path of least resistance. The breaker box.
Something else along that same path can be considered "hot" and back-feed to a point inside above the final termination sink, i.e. the earth.
Pipes and copper rods are usually used afaik but are not guaranteed.

I heard a story of some neighborhood household electrical problems in SF that were next to an electric train station. Big electricity. We are talking tens or hundreds of thousands of volts. There was an engineering ground problem. The earth in the whole vicinity became energized, which disrupted all the other grounds nearby.

When I moved into my new (to me) house last spring, mine was acting the same way. After some research I found out that the filter was SO clogged that the heat would back up into the furnace and it would auto shutdown due to high temps.. Assuming your filter is clean, make sure you are getting sufficient airflow though the system.

and then buy a WiFi enabled TStat. You can turn it on/off/adjust it from your phone... I use that feature every single day.