The whirring noise could be coming from a few places. One is from the pulley/belt drive system. A tensioner or pulley going bad, or a noisy AC compressor could all make a whirring noise that would not really be hear at idle, only when a load is placed on the engine. A power steering pump could also make this noise but a power steering noise will ususally change with steering so I dont think that is it.
The other possibility is that it is coming from the transaxle assembly. It could be a fluid pressure, solenoid, valve or torque converter problem. The symptoms are correct for this also-it happens when load is put on the motor, and after 60mph, during torque converter lockup and when the car is cruising the sound is less noticable.
The first thing for you to do is try to determine the source of the noise-pulley or trans. You can also do a visual inspection of the belt/pulley system to see if any problems or vibrations etc are evident. You should also check the transmission fluid and condition, check power steering fluid, and take notice if the AC being on or off makes any difference. Listen with the hood open and see if you can tell what direction it starts in.
Let me know what you see or find out. Then we can go on from there.
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