Nope, doesn't burn - I use mentholatum which is a bit milder. I love eucalyptus. I find it really helps my migraines too. I found a product at a health food store called Japanese oil which has a lot of eucalyptus in it and some heat and I put a bit under my nose and on my pressure points of my face and rub inton my neck.
I've put Vick's under my nose, but not in it. Doesn't it burn? :) I also rub it on the soles of my feet and wear socks to bed when I have a cold. I don't know why it works, but it does!
I use essential oils of eucalyptus, tea tree, lemon and thyme in the medicine cup of my warm-steam humidifier at night. Especially if someone has sniffles or a cough. Smells like a hall's cough drop and it works like a charm. :)
I'm a lover of vicks or mentholatum in my nose - the smell just relaxes me - must be taking back to childhood memories.
I have a lot of nasal sores and dryness there too. What helps me with that is I use a Q-Tip and dip it into a canister of Bag Balm and line my nasal passageway with that. I know it sounds crazy, but it really does help. You can find it at most drug stores.
I have RA and also mild Sjogren's which seems to flare up every time we move to a new area. The last move, four years ago, was horrendous on my eye dryness, and I ended up on Restasis for about a year and could only wear my contact lenses for short periods.
It seemed to settled down once we'd been living here a while, however I still do many of the things you list! In addition, I find it very helpful to keep a steam humidifier in my bedroom at night. It really helps alleviate the dryness in my mouth and throat (I must be a mouth breather when I sleep!) Also, before I go to bed I use a cotton swab to apply a thin coating of petroleum jelly up inside my nose. This has been a huge help in preventing dryness-related nosebleeds.
I also keep sugar-free hard candy on hand, recommended by my dentist because chronic mouth dryness can lead to tooth decay. Sugar free gum helps, too - anything to generate extra moisture in your mouth.
Hang in there!
You do have many of the symptoms. I do too (diagnosed with severely dry eyes) and have many of the same issues as you but I came up negative when tested. Lupus and RA can also cause this so getting a good blood work up to rule out all these things would be good. Good luck with the tests Dusty.