1991 Toyota Camry Engine runs rough/stalls

I've recently run into an interesting problem with my Camry. I bought a new car a few months ago, so I haven't been using the Camry much and it's mostly been sitting on the curb. It did run well though.

The problem is every time I start the car the engine will run fine (or idle) for about 5 minutes until it starts to cough and splutter and shoot out some black smoke from the tailpipe, before it stalls. Thinking it was the air flow meter, I swapped it with another one, resulting in the exact same outcome. I also just replaced the spark plugs, and while it runs better, it still coughed, spluttered, and stalled in about 5 minutes. I can start it and put on some revs, but it sounds like it misses occasionally without a pattern. Just as an experiment, I unplugged the air flow meter before starting the car and it seemed to idle better, but not like it should.

What can I do to help diagnose the problem? I'd like to avoid bringing it into the shop if possible because I plan to sell it and don't want to dump a bunch of money into it.