You may just have a very small grey water tank. I have a 26 gallon grey tank on my TC and that gets us about 5 days of doing dishes and washing hands. We (me, wife & 2 kids) usually stay at parks so we don't use the shower much. We have a 16 gallon black tank and we never seem to fill that.
How fast does your tank fill up? How many people camp with you & what do you use the tank for?
Next time you're empty, try pouring a measured amount of water (I'd start with a 5 gallon bucket) down the sink and see exactly how much water it takes to fill your grey tank totally. I am not familiar with Spert King but it might just have a small tank. If that is the case, there are drain caps with small garden hose sized caps built into them where you could connect a short length of hose to and connect the other end to a portable holding tank with a similar cap available at almost all RV stores in a variety of sizes. (I have a 15 gallon and it is not too big) This would expand your tank capacity. Do this for grey water only. black water would not go through the small garden hose. You'll have to dump the black tank using a 3" hose but it could be dumped into your portable tank as well. I took a old hose I had and made a short 2' hose that I use to dump my black or grey water into my portable tank.
Hope this helps! Let us know.
I ran into this problem. My gray water holding tank is only 11 gal. Within two days of two adults doing dishes and only one shower a piece my tank was full. Yet my black water tank wasn't even close to full. I thought of just bringing a large sized bucket with me and when the gray water ran full I was going to empty it and pour it into the black tank. As I usually camp in areas that do not have dump stations.