It sounds to me like the transmission is excessisvely worn.
If I understand the problem correctly, after a warm-up period, the car will no longer stay in higher gears. Even a small amount of pedal pressure causes it to downshift.
This is most likely a driveability symptom of worn bands inside an automatic transmission. Many reputable transmission shops will diagnose for free; just be sure they don't disassemble anything to do it or you will have to pay the full cost to replace it.
If it's really an AW-71, you can find a used one at a junkyard for a couple hundred bucks and replace it with a friend. That's what I'd do, if the diagnosis is confirmed.
As others have suggested, you might consider having the fluid flushed. Simply draining and refilling it won't do; you need to hook up a machine and do it right. This should cost between $60-80 depending on where you live.
I hope this helps. If not, you might consider giving us more of an idea of the problem --- how long does it take before it starts to happen? Also, double- or triple-check the trans model number.
As a side note, since it seems to be temperature-dependant, it's probably not the kickdown cable adjustment.
If anyone doesn't understand any of the terms in this response, you can search the board or the FAQ for them.
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