Definitely look at the fans first. The low side pressure is high but that may just be as a reaction to the higher than normal high side pressure. If you don't have properly working fans (they should come on low as soon as the a/c is turned on) what will happen is the high side pressure will get too high as the heat is not being removed from the condenser. Sounds like your fans aren't turning on low at all causing the high side pressure to rise high enough to get to the point that kicks in the high speed fan circuit. The fans kick on high but the heat already in the system is too high and the system pressure probably rises too quickly for the fans to get on-top of the job causing it all to shut off when it gets to the high pressure cut-off.
No point in trying to diagnose / fix further until you fix the basic essential components of the system. Fix the fans so they work on both low and high speed then if the a/c continues to play up diagnose further.
No point in trying to diagnose / fix further until you fix the basic essential components of the system. Fix the fans so they work on both low and high speed then if the a/c continues to play up diagnose further.
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