So i know that Steam for example uses InternetExplorer on Windows to render their shop site in Steam. And Neverwinters launcher also establishes a connection via http... so there's probably an issue on your browser, maybe a scam included a proxy? Please check your proxy entries. And maybe you can try if…
the "save browser" option will set a cookie in your browser to identify it the next time you log in. If you are not allowing cookies / delete cookies or if you are browsing in incognito mode, it will fail.