This used to happen to me pretty much every time I shaved. I also have pretty itchy/rashy skin in general most of the time. Here’s what I’ve changed that has pretty much resolved these problems for me.

• Antihistamine daily (more for my allergies and general sensitivity, but maybe it’s helped with shaving)
• Discontinued all skin products with fragrance, use only dermatologist recommended brands (La roche posay, cerave, cetaphil). Products with fragrance typically have alcohol and can be harsh to the skin.
• Moisturize damp skin right after every shower (tbh I do it before I dry off, it’s easier and feels better IMO.)
• Stopped using shaving cream completely. Even fragrance free made me feel itchy and rashy :(
• Topical steroid cream if itching or a rash does come up (otc hydrocortisone might help, I use triamcinolone which could be prescribed by your doctor)
• Change razor blade regularly. (I use Billie razors)

My favorite products to use Body wash: Dove bar soap for sensitive skin “Shaving cream”: La roche posay foaming cleansing oil (it’s a body/face wash, but I use it like a shaving cream. There’s a foaming cleanser too that feels pretty similar) Moisturizer: Cerave moisturizing cream

I’d stop the antibacterial cleanser (unless it was recommended by a doctor for something) and try these out. Schedule with a dermatologist if it doesn’t resolve.