For acne prone dry sensitive skin I would definitely recommend that you use cream cleansers that have some acne fighting ingredients. My favorite is salicylic acid it penetrates the pores to remove impurities and excess oil etc I love Neutrogenas Deep Clean Cream Cleanser. It will treat the breakouts on your chin area and keep your skin clear and moisturized. You should also be wearing a moisturizer and an SPF. Or you can use a moisturizer that has SPF. Do you use any moisturizers or SPF?
For the cleansing step i reccomend the La Roche Posay effaclair face cleanser (for oily sensitive skin). It is the only cleanser i've found so far that does a good job without over drying my face, it leaves it quite smooth and comfortable. I do think that it does need some hydration after though, especially if you have dry patches.
Regarding the acne part, I think chin acne is a hormonal imbalance sign. Your doctor might help you with that, since you might have better and longer-term results from treating the cause. It only got visibly better for me once i started addressing my PCOS and thyroid issues, both cusing skin issues.