My personal experience was that my skin went mad during peri, with pimples and blackheads like being a teenager again, but calmed down after menopause. Then the dial turned to DRY.
Skincare now needs to be focused on moisturising rather than cleansing. I've always had very sensitive skin, and was diagnosed during peri with allergies to specific chemicals that are very common in skincare products.
I don't wear make up, and only actively cleanse if I've used sunscreen, or got very sweaty or actually dirty. Then I use a very soft double muslin cloth with either warm oil, moisturiser or an extremely gentle facial soap (handmade, and the maker no longer makes it 😢). Otherwise I just wash my face with warm water and moisturise it. I never use toner - it just makes my skin even drier.
My favourite moisturisers are currently Clinique Dramatically Different - I like the gel best, but the lotion is better if I'm in a hard water area.
If I'm not in a hurry and have the time to let one product soak in and then apply another product, I like to use a hyaluronic serum, such as Clinique Moisture Surge. Simple hyaluronic serum is as nice, but I don't like the feel of the Boots equivalent.
What do you wash your hair with? Do you let the shampoo run over your face when you rinse it off? Your shampoo could be too harsh for your face.
If you live in a hard water area, I highly recommend getting a water softener. Makes a real difference to how dry and tight my skin gets. Not as good as naturally soft water, but a vast improvement on hard water. You can also wash your face and rinse your hair with bottled water, just make sure it's soft.