Burning is the one symptom most likely to be shared by people on this board... just food for thought... though everyone's clinical situation (from dry to very wet, and disease states all over the map) varies so greatly. It's a classic symptom of ocular surface disease that does not always suggest something specific as cause.
From what you describe though it reminds me a lot of the situations which "back in the old days" (i.e. 10+ years ago on the internet) we called toxic tears. Almost feels like you're allergic to your own tears, hence the relief from cold saline rinses. - What to do, what to do - that you haven't already done/aren't doing! Sigh. When you do get those mini sclerals - I know that the fit optimizing will be taking all the attention at first but I'd advise also paying close attention to how the saline itself feels on your eye with prolonged contact. (Thinking of the extreme sensitivity issue going on... triggered memories of a couple of other cases, might be worth having a chat about sometime.)
Hang in there. The answers WILL come, in time.....
The Dry Eye Zone