I've also taken a hint from the tea world and decided to just bag it!
I've had good results using a cotton bag with a draw string top to hold my ground coffee. Dump the coffee in the bag, cinch the top closed and tie an extra knot around the top of the bag with the draw string for some extra security. Then the bag goes in the coffee and hangs out. I've tried sloshing the bag around with a spoon, trying to get it saturated and all the coffee inside it wet as well as just drop & forget. I haven't noticed much difference to either method. If I do just drop it, the bag & all the coffee inside it are thoroughly saturated by the time I'm ready to drain it.
When it's time, pull the bag out, and the coffee is filtered and ready to drink. I usually hang the bag from its string by one of my kitchen cabinet knobs hanging right over the cold brew container so the grounds can drip out and not make a big mess. You can also do all of that over the sink and not worry about it. Dump out the grounds, wash the bag, and dry thoroughly for next time. I've heard of people storing the still-damp bag in the freezer so it can't grow mold or anything. I've never had any trouble with just standing the bag up in the dish drainer rack and letting it air dry.
You can find purpose-made bags if you search for "cold brew bag" on the "bookstore site". Those are usually way overpriced at around $15 each. I got "Cotton 10- by 12-Inch Muslin Bags with Drawstring" for $17 for a dozen bags, and those work really well. Bonus points is they work for brewing huge tea batches for kombucha too, but I digress...