First of all, you will want to either mow or plow under the annual rye grass before it sets seed. Otherwise, it will drop its seeds to germinate and come up again next season. Hopefully, the straw you put down doesn't have seeds of unwanted species in it. For plant recommendations, let's start with grasses since the extensive fibrous roots of grasses are very effective in holding soil in place. Native warm weather grasses will come up from seed after the weather warms in the spring. The seeds can be planted now beneath erosion control cloth (see comments below) or you can wait until early Spring. Here are some species native to your area of North Carolina that will grow in clay:
Depending on how steep your slope is, you might benefit from using some sort of erosion control blanket. The erosion-control fabric works by slowing the runoff water and allowing sediments to fall out rather than be washed down into your lake. Seeds sown under the erosion-control material grow up through the matting when they germinate. You can also insert plants into the soil by cutting through the matting. The roots of the plants that are growing through the erosion-control material anchor the soil to stop the erosion. If you use erosion-control blankets made of biodegradable material, they will eventually disappear leaving the plants to control the problem. Most nurseries carry these erosion control blankets.
You can choose wildflowers to go with the grasses from the North Carolina Recommended list of commercially available native plant species suitable for planned landscapes.
Our National Suppliers Directory can be searched for nurseries and seed companies specializing in native plants for your area.
From a quick preliminary search, I found Lumber River Native Plants in Gibson, NC with several of these grasses for sale. Mellow Marsh Farm, Inc. in Siler City, NC has several mixes of grass and wildflower seeds that would be suitable. There are more nurseries and seed companies that I didn't investigate.
Below are photos of some of the grasses from our Image Gallery: