This. I will also say if you can use commas appropriately you are better than most native English speakers, as a point of motivation. I've also had professors with quite broken English, that are very successful in their fields.
Depends on the field really. Maths? Yeah you don't need to be forming extraordinary sentences. Psychology? A bit harder.
One thing I notice from your short post:
>when listing 2 things, using 'and' to join them you don't need a comma "my grammar and punctuation is terrible" is fine
>however if you list three things you could say "apples, pears, and bananas" and use a comma, but this is preference https://en.wikipedia.org/wiki/Serial_comma (also called Oxford comma)
Here's how I would write your first section:
>I am an ESLer, how do I improve my English writing? My grammar, punctuation, and written vocabulary is terrible.