My suggestions for your consideration:
1) As far as your Mustang repeatedly dying, do not give up hope on finding a shop that can diagnose your problem(s). The art of troubleshooting older cars and fundamental skills like "Fuel-Spark-Air" and "Suck-Squeeze-Bang-Blow" is sadly being lost by a generation of technicians who cannot function without a scan tool to read diagnostic trouble codes and then look online how to swap the related sensor that is throwing the check engine light. But I digress.
That diatribe aside, I'm concerned you have basic mechanical and electrical issues with your engine that can be resolved. CHECK AROUND FOR YOUR LOCAL AUTOMOTIVE TRADE SCHOOLS AND CONTACT THEM.
They salivate at getting vehicles like yours for teaching their students. It is a golden opportunity for them to teach basic fundamentals like wet-dry engine compression tests, checking timing, checking the electrical charging system and more. You may still have to pay - but you can certainly negotiate temporarily "donating" your car for teaching purposes in exchange for favorable labor rates. Obviously, this is not your daily driver - and I sense you have more time than $$$ - so let them have at it!
2) Disregard the idea of using a 170k mile engine (and transmission) from your Taurus. The wiring harness swap alone will defeat electrical engineers. You would also in all likelihood be facing frame and body modifications (engine mounts, transmission support brackets, exhaust system, and more) that would require custom fabrication & welding. There are shops that restore Mustangs like yours - but none are going to want to take on an engine swap like this.
3) Last suggestion is to consider a 289 or 302 V-8 engine swap for your inline 6. And a conversion to a T-4 (4 speed) or T-5. This has been done countless times and there are terrific blogs by people who have done this themselves. Note I only suggest this if the Mustang is in otherwise good condition, and you have a budget of $15-20K. Do not consider any other types of swaps. And by "good condition" I mean the underside isn't rusted out. The suspension is "there" and works. The brakes work and stop the car. The interior is "there" and complete. Lights, dash, doors, windows, trunk, fuel tank, and other items like this have integrity.
There are shops who will do a V8 conversion like this (you may have to trailer it a long way to a GOOD one) - but most will reject this type of work for what I call "basket cases to start with".
Regarding my third suggestion above, while this may hurt your wallet in the near term, you would likely recoup your investment in the long term. Just look online at correctly done, well thought out, solid 1964-1966 restomods and what they are selling for. They are commanding $40-50K.
On a related note, it is never too late to learn how to do this yourself. Right now you have a non-running Mustang 6 cylinder auto that basically doesn't run. I don't know what you spent on it (and won't ask) - but I'm guessing you are upside down with respect to nt present value. That is however an option. Re-sell it now and cut your losses
If that is not what you want to do, why not take this on as a learning & once-in-a-lifetime project yourself? YOU LITERALLY HAVE ONE OF THE MOST ICONIC CARS ON THE PLANET and if you can restore it tastefully and properly yourself, you will be incredibly rewarded - both from a sense of satisfaction POV and value proposition. Take all the time you want. Study, learn, budget and spend wisely, you will make mistakes, but learn as you go.
There are dozens, if not hundreds, if not thousands, of owners who have done this and subsequently reflected it was one of their best accomplishments when they finally took their wife, best friend, child, whatever on that ride when it was all done.
Food for your thought.