I cut the vines and leave the sweets in the soil for at least a week.
We grow ours in "Totes" so they are easy to harvest (and it keeps
voles from feasting on our sweet potatoes).
Once the harvest has been sitting for the two weeks, we dump the totes and separate the sweets, we have a space between one of our dog's houses roof (pallet dog house) for curing the sweet potatoes, it's out of direct sun but has plenty of air flow.
We cure them for three weeks this way then move them to boxes for storing over winter. We usually have
enough to last until the next harvest.