I like to wear nice perfume so I gravitate towards unscented deodorant and I swear by "Dead Down Wind" they usually make products to mask your scent for hunting, but recently started to make deodorant and it works REALLY well. It survives my long gym sessions, so it holds up. But I also have sensitive skin so I don't use it daily. My go to daily is Brickell's unscented.

For bodywash I like to use bar soap, Shea Moisture is usually the brand I gravitate to since I can find it in most stores, and I like their lotion as well because it is unscented.

Full disclosure, I don't have dark skin, but I have tattoos covering most of my skin on my body and haven't noticed any sort of whitecasting when using these products. Well except the deodorant, but I kinda expect that from deodorant that isn't the gel kind.

Growing the beard is definitely the easiest option, but then you have to start looking into beard oils to help it stay nice if you grow it long. Keeping it full but not too long is the happy medium that works well.

That's a tough one though, without a local farmer's market it can be harder. Next best option is find a commercially made soap with the fewest ingredients and use that, it will be an improvement.

Good luck!