@Angelmanmom lol At least 4 or 5 decades ago when I bought my first gold rope from a local jeweler, he recommended caring for it with hot water and a dash of dish detergent. Ever since, with any washable jewelry, I've always used the same technique.
As for brand? Whatever I've been using for my dishes at the time I wanted to clean my baubles. I do like the Dawn Powerwash you're using.