|I used both the S Sandys and OS from 6.5 lbs without any leaks. By that time, my preemie's cord had fallen off, so it wasn't a concern, so I can't help with that. I honestly didn't feel that the diapers were really bulky even at that low weight, but I also wasn't trying to get him into pants, just sleepers and one-piece outfits. (In contrast, I did feel the diapers looked a bit bulky on my hefty DS#1, but he didn't fit into pants even in disposables, so I think everything looked bulky on him, because he was bulky himself!).
Someone who's used the XS Sandys (assume that's what you mean by newborn) might want to double-check my thinking here, but if you are trying to skimp on covers, how about skipping the XS covers--you can use S covers with the XS, but can't use XS covers on anything but XS diapers. The S covers will be bigger than necessary, but should smoosh down under clothes. Again, maybe someone who's done this can comment, but the bulk really comes from the diaper, not the cover, so I would think you'd still be able to benefit from the trimmer fit of the XS Sandys while having a cover that you can use on both that and the OS.