To start take a look at the active ingredients on your current cleansing products- if you can get as little alcohol as possible, perhaps try a 1.5% salicylic acid soapy cleanser and try to minimize the strokes to apply and wipe/wash it off. I rec a light moisturizer, Neutrogena has some for your face. You may still want to check with a derm to see if you have eczema.
Maybe something with oil but direct oil isn’t good for someone with sensitive skin. I’d recommend a toner w/spot application in place of an astringent as well, as they tend to have a lot of alcohol which dries skin.
So my uncle used a pin to "free" some of the hairs on my beard that may be prone to growing back into my skin.
I'm letting my hair grow out for the next 2-3 months to let my skin heal. I'll only shape up the beard, at most. In the mean time I'll research different products. Time sometimes is the best remedy.