Including keywords in your domain name is highly recommended. It's one of the easiest things you can do to climb a few spots in the SERPs.

If you decide to register a new domain name, let it age for a while before you move your site. Google likes to see domain names that have been around for a while, so making the change too soon can hurt your rankings. When and if you do move the site, you'll probably see some pretty significant drops in your search engine results for several weeks to several months.

Whatever you do, hang on to your current domain name as it's already aged and others like web directories will have linked to it. Keep in mind that if you've invested a lot of time and effort marketing your present URL, any changes will require that you do it all over again.

For now, I'd recommend that you register a new domain and have your host assign a redirect that bounces clients to your present URL.

You can have as many domain names as you like and, as long as they all resolve to one URL, it's not going to hurt. Additionally, there's always a chance that someone randomly types in your new domain name.

Having keywords in your domain name can help, but its only one of many things that can influence the ranking of your website. There are a lot of on-page things you can do to optimize your homepage to improve your ranking for "web directory". I would suggest you either start by reading an introduction to SEO or hiring an SEO. Then you should start getting links to your site with that phrase in the link text.