I also use WSJ on-line and other subscription sites, they can be difficult to set up properly. In addition to the other advice listed, if you're still having problems, look through your privacy site list for related sites that also need their security setting loosened up. For example, variations of both wsj.com and barrons.com will need adjusting to browse the site.
Like you, I don't like leaving bits of their code on my machines, so I use CCleaner to remove most of the "stuff" they leave, then manually remove more from places they hide it. I wouldn't suggest the latter unless you really know what you're doing.
Firefox is easier to use with many of the paid sites, IE can be a real pain to adjust.