I'm so sorry you're having problems (((HUGS))). Skin problems are miserable!
These are not essential oils. Palm oil, coconut oil, and castor oil are plant triglycerides and are used as anti-humectants. The essential oils I can think of in hair products are grapefruit oil, orange oil, chamomile oil, lavender oil, rosemary oil, and tea tree oil. There are others - I just can't think of them now - lol! Here's a list
of common ones.
I have a problem with products that are what I think of as "highly fragranced". I don't know why, but I can use any product on my hair/scalp with no problems, but when it gets on the rest of my skin I break out like you do. So - I do my hair in the kitchen sink and have no problems. I shower at night then do my hair every 2-3 days.
You might try using a very gentle, unscented baby wash on your skin until it clears up. Also - hydrocortisone cream really helps.
One other suggestion - you might try making your own FSG. It's great for curly hair and I haven't heard of anyone having any skin sensitivity to it.