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Today I'm noticing that when one of my window A/C's compressor kicks
on it is flicking the lights. But these lights are on a different
circuit (same panel). I did not expect this. The A/C is pulling
about 10A (120v) on a 20A dedicated circuit. Other power being drawn
from this panel includes:

So total I doubt I'm drawing more than 30A of 120V, so I can't be
maxing out the feed, right? That feed should give me 96A of 120V (80%
of 60A at 240V).

So what's the deal?

Your math is wrong. The AC is pulling many more amps momentarily during
startup, thus you get the dimming on other circuits. Very common.