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Be sure that the fuse in the main load center isn't blown and then there is a fuse in the ceiling box that you can see when you remove the small access cover. IF those are good then you can test the cable, but unplug it first from both the ceiling box AND the white receiver. The problem is probably not in the remote, but take the batteries out and leave them out for about 30 minutes AFTER you do the other checks. These are NLA and so if one of these is bad and you can't find a replacement that someone has, you will need to replace and upgrade to the newer system which doesn't use a remote. These are really scarce now! Try those things I mentioned and then get back to me if we need to go further. All follow ups are included in our fee and fully expected.
If you remove the white ceiling grill there is a silver box about 8" square up in the opening. The white receiver plugs into that box.
It would have been good to include that in the beginning............take the fuse out at the load center for the furnace and those should go off. Let it sit for 10 minutes and then put the fuse back in and see if it will reset. IF you get the same condition, see if the A/C will come on if you ask for cooling.
This is a Dometic remote we are talking about?
Was the furnace running..........or at least the fan portion?
So the furnace was actually on a putting out heat? IF so, what wasn't working here?
So do the reset as I mentioned. Take the batteries out of the remote and leave them out for at least 10 minutes. Find the fuse for the furnace and remove that for the same time. See if that will reset. If not, you have some damaged components. It will be hard to find replacements if the remote and receiver are at fault due to them being out of production for so long.
You will need to upgrade to the newer Single Zone Thermostat kit. About $90 or so and it includes a new ceiling box and digital wall thermostat.