Your tanks will go to one or two sewer connections on your Georgetown. I'm a fifth-wheeler, so I don't know your camper specifically- but my expectation is that there is only 1 sewer connection on your rig. The way that your camper is likely plumbed is that each tank goes to a pipe. Those pipes come together to a common sewer outlet/connection. Where the pipes come together are going to be a pair of valves that let you open one or the other (gray vs. black) to control what comes down and out of the pipe.
You connect a "stinky slinky
" hose to the motorhome's sewer connection on one end and the site's sewer hookup on the other. The kits (like I linked to) are nice because they give you a 90-degree hookup for the site because most stick straight up from the ground.
When you're at the site, you leave your valves closed
so that the gray water and black water collect in your tank. Once they're full (or close to it), that's when you dump them. Some people will leave their gray tank open because it can fill fairly quick with showers and whatnot. But you always want to leave your black tank closed. If you don't, the peepees run down the tub, but the "solids" get left behind leaving you the worlds worst pyramid of doom. Having lots of water per flush and waiting for the tank to be mostly full for emptying makes sure there's enough water to send all of the "stuff" down the slide to the sewer.