The new cart is stable and live, though we're still finessing a few minor usability concerns like the Continue Shopping button you mentioned, so we haven't made it the default for all new account registrations quite yet.
The yellow alert bar at the top of our Admin panel has shown a new cart notification for some time now, linking to this thread with instructions to opt-in for trying out the new cart. We won't switch any existing clients to the new cart automatically, as we have no way of knowing who may have customized the classic cart on their site, so existing users of the classic cart would need to deliberately opt-in for the new cart if they prefer. Eventually there will be a preference setting for this in Seller Admin.