Hey, I actually live on Kaua'i (but I'm away at college right now). I'm not that familiar with places to run on the southside of the island near the Marriot, but I know that the best places to run are on the road or the backroads going through the sugar cane fields. (I usually enter these roads that say "No Trespassing" because they aren't used anymore since the Koloa sugar mill is no longer there). Koloa is on the southside near Poipu.
By the way, are you staying at the Marriot that is in Poipu or the one located in Lihue?
As for running up in Koke'e, it would be a bad idea to run the trails there. They're mainly used for hiking and sometimes for biking-which is rare. They are often narrow, rugged, and sit near the edge of a cliff.
If you go up to the northshore, run up Kalihiwai ridge road from the beginning and back because it will give you a pretty long run. If you are willing to wake up very early, you should also run on the road from Hanalei beach to Haena. This is one of my favorite runs, and for part of the run you can view the Hanalei Bay. There is actually a Haena to Hanalei 8 mile race that happens every summer...(Run early on because later in the day there are more cars and the road is twisty).
Oh, and one other really great run: Go to Kealia beach which is in Kapa'a, and run north along the dirt road. (There might be construction going on at the beginning of the dirt road but you can run past it.) This run borders the ocean for the most part, and if you run for awhile, it eventually brings you to Anahola. You can even jump in the ocean at the end of your run! (The best part.)
I hope this gives you some running routes you can use.
Sorry for the long response! If you have more questions about Kaua'i just post it.
Aloha : )