I'm not sure what you're getting at here. Most carts assume that they own the whole page (and everything else), and you can't wrap other stuff around them very easily (perhaps with an iframe, but it's probably trickier than it looks). There may be a cart or two designed to cooperate better with being integrated into a larger page.
On the other hand, if accept that the cart's pages will be controlled by osC or whatever shopping cart software you choose, and the rest of the site by whatever you put into DW with just a link from DW pages into the cart (and maybe one back to your main page), is your question, "which one should I design first?" That's kind of tough, but I think you would probably be better off styling the cart's pages first. Some aspects of the layout will be hard coded or otherwise difficult to change, although most recent cart versions use CSS for much of their styling. There are also themes or templates for many carts that you might start with. After you have the cart's pages looking as you wish, you are free to lay out (using DW) the rest of your site so that it's more or less compatible with the cart. If you try doing it the other way around (laying out an arbitrary look in DW, and then trying to get a cart to match), you'll probably end up with a lot more work (if you can do it at all).
BTW, don't even think about using DW to edit cart source code (usually PHP). Learn to use a real file editor (ViM and Notepad++ both come highly recommended). Also, many carts depend on "sessions" to maintain your log-in and cart from page to page, and will forget everything once you jump out into a non-cart page. So, try to discourage navigation that takes a user out of the cart until after they've checked out or otherwise have decided that they're done with shopping.