Side question: is there a way to design this problem out of
circuits?

E.S.-

I can't say for sure about your circuit, but motorboating in general happens
when peak current in the output stage develops an AC voltage across the power
supply impedance, that is amplified by an earlier stage.

Yes, a separate regulator should fix the problem, but seems like over-kill. If
you look at existing designs, they usually feed the output stage directly from
the power source, and feed each earlier stage via a series resistor with a
parallel capacitor across the previous stage.

Values of the resistors and capacitors depend on how much voltage drop the
earlier stage can tolerate, and the lowest frequency of operation (or
motorboating).

73, Fred, K4DII