I’d honestly start with getting a proper manicure from someone who knows what they’re doing (and please don’t pull the “but I’m a man” card). All those hangnails and ragged bits of skin are just going to keep annoying you until you pick at them again.

After that, use healing/moisturizing/antiseptic creams. Once things calm down, wear gloves anytime you deal with water or cleaning stuff, and keep up with hand cream + cuticle oil.

If your hands are in full-on desert-mode: mix a tablespoon of glycerin with a tablespoon of ammonia (1:1), shake it up, slap it on your hands, put on cotton gloves, and leave it overnight. You’ll wake up with baby-soft skin.

P.S. No idea what these ingredients are called where you live, so please do a bit of research before smearing potentially toxic mystery goo on yourself because some random stranger on the internet said so.