I need to replace the ivy with something that won't choke out the others, and can be easliy contained from spreading into the lawn. The area is partly sunny to full shade. It also needs to be sturdy, as I have dog that runs through it, chasing anything that moves.

I really do not know much about ground covers so I am open to all suggestions. I have seen wild ginger at one of the local nurseries, but do not know if it would be suitable or not.

In zone 7, there are a lot of groundcovers that will fit the bill. I prefer evergreen ones so that they do their job and contribute a presence year round. Many groundcovers are rather aggressive spreaders, but that is their purpose and I've yet to encounter too many that are not rather easily managed, unlike the ivy. My personal preference is to avoid Gallium (sweet woodruff), as it grows through the roots of other plants and is difficult to manage and in my area at least, is plagued by a fungal leaf blight each season. I also avoid Aegopodium (Bishop's weed) as it is both herbaceous - disappears in winter - and is a rampant spreader.

The attached list may help. The asarum or wild ginger is an excellent choice for a more shady area but I'm not sure how well it would stand up to dog traffic. I'd consider ajuga - will transition well from sun to shade, is very durable, evergreen, offers a nice spring flower, is a good weed supression and doesn't easily get out of control.