Wow! This is new to me. Are you sure it's not just genetic, and your pores are too small? If it's clogged sebaceous glands, then I can tell you MY clogged pores are no more. I used to get grease rods that pushed their way out of my nose and face, long hard grease rods like blackheads. Since going Primal, I rarely get these, so I can vouch for positive changes in skin oil production. But, of course, I am not you. As for whether or not those changes will help your condition, I can emphatically suggest it's worth a try. Give it a month or two, the body takes it's time with changes like this.
Here's a few oil guidelines that helped my skin:
Get plenty of saturated fat, whether it's coconut oil or tallow, just make sure to eat some every day. Easiest way is to cook your veggies in it.
Don't over-do the polyunsaturated oils. A few people get worse skin from taking fish oil, and you don't need to anyway. Eating some nice fatty salmon a few nights a week is more than enough.
Monounsaturated oils seem to be neutral as long as they are coming from olives and avocados. Avoid olive oil unless you can prove it's real. Almost all olive oil is just soybean oil with lies on it. Soybean oil will destroy not only your skin but your arteries as well.
Get plenty of herbs, teas and other bitter green things. These will reduce inflammation in your skin and *might* assist your glands or pores in opening.
As a temporary measure since you probably don't know whether or not they cause you any trouble, drop nightshades. If you are susceptible to their alkaloids, you'll find out when you add them back in later.
For now, I'd recommend (and I'm not a scientist or expert, just a person with experience) very rudimentary eating, almost boring, to control variables. Red meat, sat fat, some fish, liver, fowl if you want - might want to hold off on eggs even, unless you're sure they aren't a problem - cooked and/or raw greens depending on your views, sweet potatoes etc. As a short-term experiment, these foods are relatively complete nutritionally.
Can't wait to see others' advice on this odd condition and hopes of rising above it. Hope it goes right for you!
Steak, eggs, potatoes - fruits, nuts, berries and forage. Coconut milk and potent herbs and spices. Tea instead of coffee now and teeny amounts of kelp daily. Let's see how this does! Not really had dairy much, and gut seems better for it.