I ve been working out for almost 13 years and I started when I was 16. I never wanted to use gloves as I wanted my palms skin to get tougher and I never liked the feeling of the gloves when working out.
I had the same problem when doing pullups so I tried many grips so I can find the one with the least amount of pain specially that I ve got heavier.
Honestly didn't find anything that helped as much as to focus on my back and mentally ignoring my palms pain. I did not count the reps and never passed 15 reps in a single set as far as I remember but combining pullups with deadlifts, t-bar rows.. gave me a wider lats and a stronger grip.
What I do also to make my palms more appealing at work or around my GF is to rub my palms with vaseline before I go to bed and to trim from time to time the bumps in my palms so they don't get bigger.
I hope that can help.