@lilmissy don't harvest the sweet potatoes until the green foliage on top of the soil starts dying off and then do it.
The secret is when you harvest them to keep the dirt on them and trim off the side shoots leaving a bit attached which keeps them for ages. We just place them on newspaper so they are separated with the dirt on on our veranda with shade cloth to keep the sun off them. Bear in mind you would be wise to use the smallest ones first or blanch and freeze them, but the bigger ones will last a lot longer just kept like this.
The curing time just makes them sweeter.