Everyone is different but for me any kind of processed oil (natural, organic or otherwise, including margarine, butter, vegetable oil, coconut oil, etc) has not been good for either my digestion nor my skin. I prefer to get my oils in their natural state...whole and part of the foods they originate in --- i.e. eat wild salmon versus taking an omega-3 pill, eat whole coconut versus processed coconut oil, eat whole olives versus olive oil, etc. I think the same thing can be said for oil as has been promoted for other foods...it is far healthier to eat whole foods versus processed. Why? Because there may be factors present in the whole food that help with digestion (enzymes) and assimilation that are lost when the whole food is broken up and parts extracted.
I had terrible digestive problems until I started avoiding all processed oils. Now I eat lots of wild fish, free range fowl, avocados, olives and other high fat foods in their whole and natural state and my digestion and elimination (sorry) is perfect. And my skin has benefited too. It took a couple of weeks to get the benefits but it was dramatic enough that I have no problem following this way of eating --- which is really more like how mankind traditionally ate!
You mention drinking coffee. Coffee is a very concentrated drink (ground up roasted beans) and may be too harsh for a sensitive stomach due to naturally occurring acids. If I might suggest an alternative. I have found roasted dandelion root tea to be a tasty alternative. It is especially nice with soymilk. Plus dandelion root is good for inflammatory conditions. You can find it in the health food store. It has been used in Eastern European countries for centuries as a coffee substitute.