Flakiness and the hot shower sounds like your skin is really dehydrated. BP is really drying. Read more about dehydrated skin here.
I would stop using BP for now and first get your skin hydrated. If you're not using a moisturizer at all, this may be all you need.
After washing your face, use a moisturizer for your skin type. Oily skin- get an oil free moisturizer. Don't use an AM moisturizer with sunscreen at night because the sunscreen will clog your pores. If your face feels oily after using a moisturizer, then it is too heavy for you and you need a thinner moisturizer.
If your moisturizer is too thin and not hydrating enough, try adding a very thin layer of vaseline (or any occlusive) on top of your moisturizer to seal in moisture. Look up using occlusives on skin if you're hesitant.
Remember, don't use a lot of product. You generally only need minimal amounts of product in skincare.
Then slowly add BP back in, maybe once every other night. And always moisturize afterward.
Don't use too hot or too cold of water, don't rub your face, and I would stop using a washcloth. These are all harsh, irritating, and drying to your skin-- especially in combo with BP. Rub it on your hands (as vigorous as you want), then gently gently gently wash/apply on your face. Pat your face dry with a washcloth, never rub.
Exfoliation removes dead skin cells, with either physical/manual exfoliation (think clarisonic), or chemical exfoliation with AHAs & BHA (Salicylic Acid). More info here at r/SkincareAddiction. The BP is not a chemical exfoliant, so it does not really act to remove dead skin cells.
I would advise working anything new into your routine slowly (nighttime, once every couple of days at first, then work your way down to every night) and only after your skin is hydrated.
Listen to your skin. If your skin is sensitive to every night with BP, but you get good results every other night, then stick to every other night.
IMO I like chemical exfoliation better than BP for skincare because it is hydrating and can also fade scarring, but I'm just a person on the internet. Read up and decide. If BP is working for you, continue on.
I hope this helps you!