the signals when changing from one rail to the other:

So it seems once its has decayed enough so that the top of the 2nd-integral signal no longer clips, then that triggers the resumption of proper integration...but what happens is the decay keeps on happening and actually over shoots and hits the other rail.

To lessen likelihood of entering this situation, should always DC block the input signal.

zoomed out a bit:

If instead did two RC low pass filters in series, then I'd think at least the "integration" would always remain around the middle voltage, so wouldn't have this rail hitting issue. The capacitors would always be keeping the proper voltage. Would still need an opamp afterwards to amplify the signal so it is the same amplitude range as the input signal.

but if try two RC integrations, then get not so great frequency spectrum:

There is a 3rd harmonic. So the question I need to investigate is can I get rid of that harmonic with RC and just deal with RC-only, without having to even involve opamp? Or is the issue something else that can be avoided in some other manner?

the pwm seems to be close to 50% during rail transitions:

ADC 2nd integration (yellow) also hits the rails