@Anatoly, you might want to invest in a set of rings - these will let you ring a comfortable groove for pullup, chinups, and all the in-between positions. You might also want to find a pullup bar with neutral position, where your palms are sideway and facing each other. I have these, amazon.com/gp/product/B01KHPSXJM, bought 3 years ago and still holding up well for me.
Another thing would be to get into a chinup grip on the bar while standing on a box or otherwise positioning yourself to take some of the weight off your hands/wrists, and you can experiment with easing into adding more weight.
And yet another choice is one hand over, one hand under.
-S-