I've had ultra-sensitive skin all my life, and as an adult I have Rosacea. The things people like me can usually use (like Clinique cleanser or makeup, mineral makeup and Cetaphil) make my skin burn, even to blistering with Cetaphil.
The best cleanser I've ever used, hands down, has to be Pond's Cold Cream. I don't know why - it violates just about every rule there is - but it's awesome for me. I even use it as a moisturizer before bed. I can also use Arbonne's RE9 night creme, but it's FAR more expensive. I use Philosophy's Hope in a Jar and The Present under my makeup because the Pond's is too heavy for daytime.
I can't go out without full makeup because I need the physical barrier between my skin and the sun, as well as against environmental pollutants (and there is no way I can put any kind of sunscreen on my face). Strangely Lancome has been the best liquid foundation and powder for me. I'm currently using Revlon ColorStay because they didn't have my color at Lancome last time, and so far, so good, as long as I take it off (with the Pond's) as soon as I get home.
You have to try, try and try again until you find what works best for you. Certainly try to see a dermatologist, but it will probably take a while to get an appointment. In the mean time, just experiment and see.
I'm sorry you have to go through this.