I would start with a quarter turn out on the barrel adj. and then check cable tension and cable friction. Per OP It shifts to a larger cog and stays put but when shifting to a smaller cog it jumps an extra one under load. I experienced that when climbing hills and it only took a quarter turn out on the barrel to get it right. Not saying anyone is wrong here, but that is a cheap and easy couple of things to eliminate before buying parts and throwing money at it.