Never use design view in Dreamweaver. It’s not a browser and it’s more wysiwtf than wysiwyg.
Just use the code editor (which is very good) and code by hand and then view the result in a real browser (or two or three browsers).
I usually load the page I’m working on in three browsers (IE, Firefox and Chrome) and as soon as I change the css or html (or at regular intervals) I just click refresh in each browser to see the result. That’s the most reliable way and also does most of your browser testing while you design. Don’t use the F12 preview in DW either as that is also occasionally broken and sometimes doesn’t give true results.
That’s the way I work anyway.