The only DOMS I have experienced of any note is when the day after a hard workout (first time back doing squats for years), even though experiencing muscle soreness, I followed with my usual 4.5 mile walk, which left me practically unable to walk at all.
Exercising when experiencing muscle soreness results in longer recovery time from that soreness or increased soreness. It never results in any sort of improvement for me.
In the end I decided to not exercise the same body part again until the soreness has gone away.
This has eliminated DOMS, too. Imagine that!
Pushing through the pain is for people who give themselves points for enduring pain.