I was a plumber in San Francisco for 20+ years; my sister and I own two 100+ year old houses in the Chicago area. Have "done houses" in every way all my life and in 2008 replaced the control unit, various sensors, pumps, tanks, valves and stuff in a 40 year old AOSmith copper boiler heating one of our houses (hydronic). I tend to look at fixing things before buying more. That said, I'm as fallible as the next person, I just have more practice... <g>
Your furnace does not sound like it is necessarily dead/unrepairable.
But. We need to work with the resources available and for you that sounds like it means the
local contractors of various types and persuasions, so before talking about your furnace I'll run through briefly my standard text on contractors. It's not pretty, but it's the best I've found. Most building contractors find it quicker/easier/more$ to just replace stuff, but there are some that take a more nuanced approach. Usually it's far far better to call for repair and listen to whatever music they play than to call in saying you need to replace your <whatever>. Let _them_ tell you what's the problem and what they can do. Then ask questions, take their info and quotes and move on to the next prospect. TAKE NOTES!!! It will all mush together, I promise. Make a spreadsheet of your prospects including your first impressions and your last comments.
It's a large pain, but with luck it can be educational. I don't know any other way to give yourself a chance of finding people good for you. When I need a new contractor, I have in the past interviewed more than a dozen and then had to pick from two or three that seemed somewhat plausible. Location of the shop matters somewhat, although contractors do travel 5-20 miles (depending on roads). But provided other things being equal, closer is better because it makes it easier for the contractor and it makes it easier for you to go to personally express yourself if needed. Remember: The relationship will never get any better than it starts out. You need people who SHOW UP ON TIME (or call well before not showing), that you can work with, who can work with you, who you respect and who respect you. Listen to your feelings, talk to some of them again with questions, changes... See how things play out and record it in the spreadsheet. Pick one and go for it.
Ok, about the furnace. It sounds like some kind of electrical problem and those can always be fixed, even if it means replacing the "motherboard". That doesn't automatically make it the best go, but it's an option. There may be more than one problem; important things may be corroding, something else may have _caused_ that electrical problem. Listen to opinions, look at what they (
should) show you, talk with the next guy. It could even possibly be the super smart thermostat having a hairball. One thing that would absolutely require a new furnace would be cracks in the firebox that could let combustion products mix with the air sent into the house. If that is found, have them show you the problem area. If they can't they better have a _very_ good explanation of the whole thing and then you get a 2nd opinion (unless you just want a new furnace...).
Your ducts are not a no-brainer. They may be ok, they may be leaking and corroded to hell. They are almost certainly not "good" because most ducts in this country were never installed "good". But if it has worked ok for you until the furnace got your attention, you can probably go with the evil unto the day and leave the ducts well
enough alone until you are given more motivation.
Finally, I often refer people to "heatinghelp.com". Its forum, "The Wall", is mostly heating professionals and they mostly do hydronic heating. However, my
experience has been that if you do your homework before asking questions, stay on topic and act respectful, they will try to help out regardless of your system type. It would probably be worth some reading time on their site just to get a feel for that type of work, even if you don't message with them. One of their categories on the right of the screen includes heat pumps and you could find some informative posts there. They also have a "find a professional" page and if you don't find any leads, ask under their "Main" topic about furnace contractors in your area.
Regards,
Rufus