@Susan - Congratulations on your initial, very successful year. Alder is exactly right, of course: if you are primarily interested in a root harvest, then you should be happy that yours were growing so tightly concentrated beneath the original plant. How convenient! In my own past experience, that wasn't so much the case: the more I dug, the more I found sweet potatoes hiding in all corners at all depths. But there could be your
answer: harvest only the most convenient and obvious roots nearer the surface around your original plant, assume that there are more scattered further afield, and see if they reappear on their own early next summer. If you don't see sweet potato vines by a certain date the second year, then assume that any left in the ground have rotted over the winter (all the better for your soil!), and plant new slips. If you save a few of your own sweet potatoes over winter for growing new slips, fine. If you buy new slips instead, fine too; they're not expensive. In the past, I grew slips by taking whole sweet potatoes, laying them out on top of a couple inches of straw, covering them with a couple more inches of straw, and watering them occasionally during the very late spring. By early summer, as the first vines poked through the straw, I just buried the whole sprouted sweet potatoes in my garden. This produced vigorous plants and large harvests in my own case. But of course, you could pick off the new slips and plant them individually, as Alder described above.
In this scenario, you wont be doing any more digging than what was necessary during your fall harvest, at least not until your drop-dead date has passed the following summer and you go to plant new slips of your own. Before that, you don't want to disturb anything just in case your prior years root system has gone perennial and is preparing to break surface again, which from my perspective is the ideal outcome. Worth a few year's experiments, at least, to see if it is possible in your climate. The ideal goal of perennialized sweet potatoes is ideally suited to a no-till system (which is best approach anyway, in my opinion).
As for your disappointing storage results, Alder is of course correct on that point too: sweet potatoes, like winter squash, appreciate warm dry storage at room temperature or just below. But there is another consideration, as well. During the fall harvest season, there is a delicate dance to be danced. You want to wait as late as possible so that your vines grow the biggest possible tubers. But, if you wait until there is significant, visible frost-kill of the vines, the sweet potatoes you dig will be compromised. They will look and taste fine when first harvested, but their long-term storage potential is reduced. At least, so I have read.
@Alder - You wrote "All of these should keep well into spring if cured and stored properly..." I know what you mean about curing winter squash: after harvesting, they are left sitting in the sun for some extra days (up to a week?) before storage, so that their skins harden. Do you do the same with sweet potatoes? I didn't know that. Inquiring minds want to know...