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The behavior of your air conditioner is consistent with accidently switching the heat mode on.
On through wall units like yours that is typically set with the mode switch on the remote.
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You mean dim the lights for a moment? (in an LED light, this can appear as a flicker or a bright flash)
That is more of an indication of crappy wiring then of any issue with the AC. However, we can glean a lot of information from it. Specifically, that much current draw indicates mechanically the fan is spinning up and the compressor is engaging. Your capacitor is fine if the fan is spinning, it is just used to put the one winding out of phase so the motor can start. Capacitors do not increase current draw as they fail in this application. Nor will the capacitor increase voltage on the line.
If it was just the fan engaging it would likely not pull enough juice to effect the lights even on a poorly wired circuit. So likely your compressor and fan are engaging. We can rule out control board / electronic failure also.
The bad news is odds are you are either:
out of refrigerant
your compressor has failed (internal seals etc) and is no longer pressurizing the coolant