True natural dye black is hard to achieve, and to dye it onto cotton is harder still; cotton is a woody fibre and though it stains easily, it doesn't dye easily (stain is just a grubby non colour, a dye is a deliberately achieved tone or colour)
The oak galls are rich in tannin, but you can use oak bark just as well. It's the interaction of the oak and the iron that gives the black. Too much iron though rots fabric.
Some of the American dyes are better for black than ours; logwood can be used to good effect while most of our blacks are green/black or rusty red/black or charcoal grey.
Tbh the only way I get a good black natural dye is to mix the materials. Oak galls ground to a powder, iron mordant, green walnut husks and iris roots, give me a black. It can be a bit hit or miss though, and can end up a charcoal grey black,
I wouldn't even attempt a big bit of cotton cloth, it would need to be kept moving in plenty of liquid and kept at least at blood heat temperature. Just too much bother for me tbh.
I got a better effect by adding indigo but that wasn't stuff I'd grown or gathered. My woad blue is too pale to make much difference to the dye.
Sorry to sound so discouraging, but black's not as easy as it seems. I note loads of lists of dyes on the internet giving the same lists of suitable plants; and I *know* that those folks haven't all tried them, that they're just copy and paste lists they've lifted.
Once someone writes that 'suchandsuch' gives 'thiscolour' everyone just adds it to a list instead of actually trialing it first
Best of luck if you do have a go at it though

and I'd really like to hear how you get on with it
If I *had* to dye the cotton black I'd buy one of the wash in dyes for the washing machine. Dylon pouches are very, very good
atb,
Toddy