The Arcana mod has a tool/weapon called the “Silver Athame”. When activated, you get a message that you used it to cut yourself, and one of your body parts begins bleeding. You can then use rags to try to stop it and every attempt, whether succeeding or failing, yields a bloody rag. Occasionally, you can forage a bloody rag as well, but those are too rare to be terribly useful. I think there’s a way to make the Athame, but I’ve never done so. They are not terribly uncommon in the Arcana locations, though. There are also arcanist basements in the mod, so if you can’t find one of the locations, you can always keep checking basements. I’ve also seen Necromancer zeds and maybe Masters drop them, as well.
Edit: You can still kill yourself bleeding like this, so don’t overdo it.