Update time:
Replacing the primary and secondary o2 sensors appears to have resolved the problem. I replaced them over the weekend and the car has not given me a CEL of any kind since. I also replaced all 6 spark plugs as they were due for service as well, 75k miles.
Results from Torque:
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Everything seems to be working good on first start after everything was done.
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This is me driving around. The hills and valleys are shift points.
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And the car coming to a stop.
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Slowing down, accelerating, then slowing down again.
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Yes, I replaced the catalytic converters myself last time. I bought them from one of my local Ford dealerships and did the work myself.
Buying CATs from RockAuto, that's the first place I go to for all my vehicles' mechanical needs. I'm in California. The problem there is that the CATs that RockAuto has available are all EPA certified, and me being in California means that they need to be CARB certified, not EPA certified. That said, I cannot buy CATs for my Taurus from RockAuto, they have to be from Ford.
The lesson I learned about this is to replace the o2 sensors first if this P0420 and / or P0430 CEL happens again. Particularly if the CEL turns off and on as it was doing this time. The time I replaced the CATs (as referenced in my OP), the CEL was always on.
And no, there's no other issues with the motor.