I suspect the problem crops up more when used in longer barrels such as rifles and shotguns...not so much in pistol length barrels.
Yes, the snake should not be pulled through the barrel unlubricated unless absolutely necessary . It doesn't need to be soaked, and infact probably shouldn't be. I use a bit of bore cleaner in front and behind the bristles and depending on how much oil I want in the barrel I'll put extra cleaner at the end of the snake to coat the barrel for storage. If im planning on firing the gun I use much less oil. And I use the snake until it is just a very VERY thin layer of oil in the bore.Could it be that the bore snakes need to be really wet to work good and not get stuck in the barrel. Maybe a good dose of Hoppe's or other bore cleaner from the front to the end of the brissels and a good dose of clp or other treatment from the brissels to the end of the snake ? That way the snake is well lubricated ?