Keeps shutting off?

The vehicle listed above is a GS model, with a 5 Speed manual (the lowest model, least featured version).
It started when driving. Car would shut off while in motion (slow down, pushing the gas in did nothing). I would push the clutch in / out (popped the clutch while moving basically), could accelerate again. That worked around 10 times. Towe truck home. The next day I went out and it started immediately, ran (idles fine) for around 5 minutes, shut off. Tried to restart, cranks over fine, but does not start. Tried again around 4 hours later. Started right up perfectly. Ran (idles fine) for around 5 minutes, shut off. Tried to restart, cranks over fine, but does not start.
When running, I push the gas in, it hesitates slightly before revving up. The hesitation increases the longer the car runs. Then it dies. The process is repeatable.
Put some Lucas Gas treatment in it. Went through the process above many times. No help. The gas "should" be good.
The next day. Started it up. Immediately drove it down the block, on the way back it died (3–4-minute run time; 40F outside, cold). Light car, pushed it home.
The water hose is a little warm, not hot.
Exhaust gets "slightly" more gassy smelling and idles a little rough 30 seconds before it shuts off. If you push the gas in during that time it immediately shuts off.
Next step, wait. (See above)
My plan is to replace the parts myself until it is fixed.
No big hurry.
This will be a hobby this summer.
Price is not the main concern. I would like to start with the easiest fixes. Depending on cost I would likely order many parts, just replace lots of stuff in a weekend. See if it is fixed? If not, replace the next batch of parts.