Can my car drive 350 miles with this issue?
There are several possibilities for the issues you've described, but I would look more towards a belt being the issue (https://www.yourmechanic.com/services/serpentine-drive-belt-replacement). It could be the belt, belt tensioner, air conditioning compressor, the alternator, or one of the other belt driven...
What else could cause power steering problems ...
What else could cause power steering problems