First... Try fresh batteries.
Next (if you are a handy person), shut off furnace at breaker or 'emergency' switch and inspect the connections.
Carefully remove the thermostat from the wall to expose the wire connections. It's possible that something wasn't tightened and an end fell out or broke due to normal household vibration. A loose or detached end might also be short-circuiting another connector causing a continuous 'on' situation. The dip switches could have been disturbed when you changed the batteries but those are usually covered. Definitely worth checking as suggested above.
If that doesn't resolve the situation, check the connections on the furnace. Follow the T-Stat wire to find them. Note: Many newer furnaces are digitally controlled. If you have any doubts here you should leave it to a pro.
Thermostats are relatively cheap. Replacing them is not that difficult so, except for the time, it's not a terrible idea to replace that component if everything else looks good.
I've only once tried to help someone with a situation like yours. It was an older boiler and the issue ended up being further into the guts than I was willing to dig so we called a tech to resolve it.
You can probably mickey-mouse an attic thermostat into the emergency switch but a better idea will be to winterize the place until you can get a qualified tech to look at it of these simple checks don't work. I don't think it's worth any risks for a place you don't have to live in.
Good luck!
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