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The Check Transaxle light will flash if the transmission fluid temperature (TFT) sensor reads an over temp condition (>300 degrees F), or due to other sensor inputs. Check the transmission fluid condition to see if it seems burnt (strong odor, discoloration). If so, you'll want to get the fluid and filter changed. There are varying opinions on the benefits of flushing the system. If the system has not been well maintained, the general concensus seems to be to avoid flushing the system.
There are multiple solenoids in the transmission that could be sticking and not working as intended which could cause the flashing light.
You could also have an indication of slip based on the input shaft speed sensor to output shaft speed sensor ratio. This could be due to a faulty sensor or a slipping transmission or, as jlb1000 suggested, a torque converter that is not locking up when requested.
Your best bet would be to watch someone connect their professional diagnostic tool to the OBD-II port and scan the transmission computer for codes. The tool should explain what the codes are, and you can report back here with the codes for further suggestions.
Did the engine speed (rpms) change when the light started flashing? At highway speeds, what was the engine speed at a set cruising speed (ie: 2500 rpm at 65 mph)?
There is an electronic tie between the A/C clutch and the transmission. Per the service manual, "Air Conditioning A/C Clutch An electromagnetic clutch is energized when the clutch cycling pressure switch closes. The switch is located on the suction accumulator/drier. The closing of the switch completes the circuit to the clutch and draws it into engagement with the compressor driveshaft. When the A/C is engaged, electronic pressure control (EPC) pressure is adjusted to compensate for the additional load on the engine." This EPC is controlled by a solenoid, so you may have an issue with the solenoid or associated regulator valve.
-Rod