Yes, they dump just like a travel trailer at any waste dump. Usually 30 - 45 gallons black and 30-45 gallon gray which is plenty. One very important thing you did not mention is the fresh water tank. A lot of LQ's trailers only have 30 gallons of water. That is nowhere near enough even for a few days. Try to get 90 gallons if you can. A lot of trailers are 60 gallon, but some are 90. I can tell you there is a big difference in a 48 and a 53'er. It is nest to impossible to get in and out of a lot of places with a 53' trailer. The rear is so far behind the axles it drags on the slightest incline. Places as simple as trucks stops can be a pain and don't even attempt most small gas stations. If you have a crew cab, long bed truck a 53'er is to long in a lot of states. It'll be over the 65' max length. I know my dually is with a 53'er.
Make sure to get one with a 20 to 25 gallon built fuel tank for the generator. A smaller tank is a pain to have to refill and keep extra gas cans for. Make sure to get at least 7,000lb axles and 16" tires.
Do yourself a favor and get the right one the first time! You will thank me later!!!!