Sweet potatoes are an annual crop, if you were to try and over winter slips for next spring you would need to pot them up at some point in the winter or they would die.
The fact that those in the ground were putting out slips means you didn't dig them early
enough.
We cure our sweets for three weeks and they last us into the next year, we have not had to buy any this year and just recently ate our last of last years crop.
You can cut back the vines of sweet potatoes all year long, if you don't want to bury parts of the vine to produce more tubers. The leaves are edible as Tobias mentioned.
You can also make new plants by cutting and rooting parts of the vines.
Sweet potatoes do require water during the growing season or they will not be as good as they can be.
If your slips died upon plant out, they either didn't have enough root growth at planting time to survive or there wasn't enough water supplied for them to become established and thrive.
Your idea of potting up some up and growing them indoors is a great way to grow through the winter and then plant out in the spring.
To keep vines under control, coil them up as they get long, that way they won't take up too much space.
Redhawk