1997 Ford Thunderbird shudders and stumbles

1997 Ford Thunderbird mileage: 43,000. Last August a small shudder showed up when the car shifted into third gear (automatic) and the engine was under a slight uphill load. This only occurred when I was very lightly pressing on the gas pedal. If I accelerate even a small amount, the shudder goes away. Over the past few months it has gotten worse, meaning more frequent. But I have noticed that it never happens when the engine is cold. It has to warm up for a while. I have had it to a transmission shop that comes highly recommended. They say there is nothing wrong with my transmission or torch converter. I have also replaced the spark plugs and the plug wires. There is one more observation worth noting. When it does occur and I can keep it in the 'shudder mode', it sort of cycles in and out of the mode, cyclically and very periodically like it is part of a failing servo loop or something. If I were to guess, I would say it’s like the fuel to the engine is being modulated like the system erroneously thinks, the engine is running to rich so it leans the mixture, which causes the engine to stumble slightly. This stumble (shudder) causes the system to richen the mixture and the engine pulls harder, then the process starts over again a few moments later. I have thought about seeing if I could find the oxygen sensor and unplugging it to see if that would affect the problem.