Directories give you result in delay but it is a good practice for the future. I suggest you to submit your website in DMOZ first and other search engines like Google, Yahoo, Bing etc. Add live chat to your website to interact with website visitors.
The most easy way to search web design domain is to land in GoDaddy website and put your desired domain name in search domain field. There is more options to select domain name if available. GoDaddy also shows premium domain names which are on showcase.