On a few models of campers, the bathroom sink will go in to the black tank. Never seen a shower go in the black tank though, not even safe to do so.
Now you said when you turn on the bathroom faucets, the black tank fills up. Would take several hours to get enough water through the sink to fill up the black tank. How do you know it is filling the black tank? Are you looking down through the toilet, or watching a monitor panel? Are you sure you don't have two gray tanks? (usually found where there is a kitchen on one end, and a bath on the other).
2012 Ford F250 ext. Cab 4x4
2002 Jeep Wrangler Sahara
2008 Yamaha V-Star 650 Classic
2008 Work and Play 18LT