Hi and welcome!
I used a series of lactic acid peels myself. Initially, my skin was as smooth as a baby's bottom, but then over time I developed skin that was as dry as parchment paper. There is the danger with peels that they will disrupt the acid mantel and the skin will no longer be able to exfoliate properly. I'm not a dermatologist, obviously, so I don't know exactly how skin functioning works. However, I know that if you wash or exfoliate your face too much you will end up with drier skin. I'd recommend using a very gentle cleanser and then dab your face with a smidgen of jojoba oil (not too much, as this will prevent your body from producing its own oil). It's important to allow your skin to breathe so it can produce more sebum and get back into balance. Avoid mineral oil based lotions, since they will only clog the pores and cause the problem to actually get worse over time. You will notice that your skin will be more dry and flaky initially when you stop exfoliating. This is normal, as your skin tries to get back into balance. You can use a gentle micro fiber cloth if the dry skin is driving you crazy. I like the one by Jane Iredale.
Also, make sure that you're getting a lot of essential fatty acids in your diet, e.g. fish oil, flax seeds, hemp seeds, chia seeds. Incorporate lots of antioxidants into your diet in the form of low glycemic fruits (apples, berries, pears) and vegetables (kale, chard, bok choy). Avoid junk food, pre-packaged foods, breads, cereals or at least eat them in moderation. Opt for less refined and more nutrient-dense grains (e.g. quinoa, amaranth, buckwheat, spelt). Drink 1.5-2 liters of filtered water a day. Practice relaxation, whether in the form of yoga, meditation, dancing or whatever makes you really happy. You might find that 1 or 2 of these suggestions do the trick or it might take some time to get to the bottom of what is causing the dry skin, but you will find an answer.