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That, of course, leads to pump starvation and the eventual demise of the pump and, possibly, the steering rack. When servicing a late-model Chrysler power steering system, the carmaker recommends using genuine Mopar fluid or an equivalent. In a pinch, you can also use ATF+4 transmission fluid, with no ill effects.
You drove 60 miles with the engine oil in the transmission, so I doubt you did much damage if any. ATF has friction modifiers in it and SAE 5W-30 doesn't. Without those friction modifiers, the clutch packs in an auto transmission won't engage with the same amount of grip which would explain the difficulty shifting.