Sweet potatoes aren't in the same family as traditional potatoes, and are not nightshades (traditional potatoes are in the family Solanaceae, while sweet potatoes are in the family Convolvulaceae). So their shoots, flowers, and roots are not poisonous generally speaking, not like a starchy Russet potato (eat the stalks of those, and you will be seriously in a bad way). Indeed, many cultures eat the stalks and flowers of the sweet potato (which is actually a morning glory). Now, having said that, some morning glory related genus and species plants do have some alkaloids, even including (yes) LSD as a naturally occurring agent, particularly in the seeds - but not sweet potatoes. Still, the creeping roots that come out of the tuber are kind of gross, so I wouldn't eat them:) But the tubers should keep for several months if kept dry and free of moisture - just carve out the root hairs like you would a normal potato eye - problem solved!
Edit: Oh, and as a guy, I would mention that sweet potatoes also contain a serious dosing of phytoestrogens - good for the ladies (recent studies show up to a 20% increase in blood estrogen levels on a sweet potato diet: http://www.jacn.org/cgi/content/abstract/24/4/235 ), but I would keep them in moderation for a man.