Along with the Calgon add some Dawn Dishwashing Liquid. Calgon may be difficult to find. I get mine online.
Another school of dumping thought is to dump as soon as you get to the new campground. The thinking is that during the travel all of the parts and pieces with be sloshed around and more likely to be dumped than if the rig is just sitting at a site for a few days.
Hopefully you have a clean connection at the RV discharge so you can see when the water runs more clear. It may take several `dumps and refills' before it is.