I also have sensitive skin and use very few products with synthetic ingredients or fragrances. For the day-to-day I use natural cold-milled vegetable oil soap with essential oils that are hand crafted by a woman in my neighborhood with an exfoliating bamboo wash cloth (anti-microbial and anti-mold). While I am aware that you cannot get the same soaps, you can probably find something comparable a health food store, which is also where you will find natural exfoliating wash cloths/hand mits.
On the days were I really want to treat myself and do a good exfoliation, I mix jojoba oil and sugar and give myself a nice scrub down. Jojoba is my favorite oil, but hemp oil also works well for my skin. Some people love shea and cocoa butters but they are too heavy for my skin and don't absorb properly. I hope that helps!
If you want to just buy a nice product from Sephora, I have been really liking the Fresh line of body products.
Just mix a table spoon of salt into the amount of body wash you use to bathe with. It will gently exfoliate without damaging skin or causing a breakout. Wash as you normally would. Then rinse.
I agree with dianabt about getting a loofa, that would be your best bet espeacially since you have sensitive skin. I use the Soap & Glory's "the scrub of your life" but i'm not sure how that will go with your sensitive skin! But it just smells amazing!
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I agree that a body loofah or sponge can do wonders in exfoliating the body and removing dead skin cells for soft and smooth skin!
I also recommend that you try a body scrub, the most affordable we carry is the Sephora Collection Sugar Body Scrub ($18 USD)
What is great with this scrub is that it uses sugar to exfoliate which is very hydrating and soothing as compared to a salt based exfoliator. I hope this can help!