Dear Delilah, welcome to the forum.
It's proteins that cause most of our sunflower allergies. I have it too. It's pretty common in areas where sunflowers are grown as a crop.
From your symptoms, it sounds more like something you are ingesting than something you are using externally.
I avoid anything to do with sunflowers. There are plenty of other products without them. The oil, if processed very well, is supposed to not be allergenic, but I don't trust it.
In general, the fewer chemicals and additives and processing, the better.
The best way of diagnosing your sensitivity is keeping a good food (and in your case, skin products) diary. There are loads of posts on food journals, diaries, logs on the HW forum, so I won't repeat. Do a search using the search box at the top of the page.