Many camp grounds we have encountered have WASTE 'GREY' WATER dump stations scattered around their camp sites. One at CloudLand Canyon in GA has two in each camp road...

Some camp sites has an outside wash up tub at the bath room buildings for washing dishes and other cleanup items to keep from having to use your grey tanks...
When using my OFF-ROAD POPUP setup I always capture the sink water and tote to the bath rooms using my two wheeled folding cart and pour down the toilets...

I have heard from some campers that they carry an empty tote in the bed of their truck with capacity size of their largest tank and will use a FLOJET Macerator Pump to transfer waste water to the portable tank and haul off to the local dump site
I guess there is several ways to address this problem...
Leaving the camp ground because of full tanks would not be my solution for sure...
Roy Ken