My parents, born in India in late 50s, rarely had access to soaps and detergents growing up in the countryside. They are still very conscientious in how much detergent they use. It's a lot of bad habits that mega-corps have tried to push onto us.
And it can go under so many names after it's been processed by the Oleochemical industry - https://www.ran.org/the-understory/palm_oil_s_dirty_secret_t...
Not eating Oreo filling or Nutella is just generally good health advice (if you want chocolate + hazelnut spread, you can buy ones made out of hazelnuts and chocolate instead of mostly sugar and palm oil; plenty of available packaged cookies have little or no vegetable oil). Plenty of available peanut butter has nothing but “peanuts” as an ingredient. Make your own pie crust using flour and butter (or lard). Etc.
Avoiding palm oil in soap/shampoo is harder; best first step is probably to reduce soap consumption. Most people use way more soap than necessary, way more often than necessary.
Their liquid soap doesn't contain palm oil.
I avoid palm oil in foods for the simple reason I don't consider anything made with palm oil to be a "whole" food. But I've never known what to do about other products like soap.
P.S. To bystanders, you may think your soaps don't contain palm oil. Check for anything like "lauryl sulfate". That's (probably) palm oil.