No need for a new thread and the problem is the transistor, not LM358. Charge storage is always a thing, not just in saturation.
I think a MOSFET may perform better, but frankly I have never investigated this case myself so it's just a guess.
A diode between base and emitter should also help, by allowing LM358 to actively pull the emitter down. This may or may not require negative supply for best results, try it and see it.
edit
I guess you could also use a PNP emitter follower instead of a diode. Base to Q1 base, emitter to Q1 emitter, collector to ground or some negative rail.