I want to attract customers who are more concerned about quality and accountability than price.
You're not going to find those customers on a freelance website or Craigslist. Ask yourself, if you were in their position, how would you find a quality software developer? Most likely you'd ask people whose opinion you trusted (friends, colleagues, or leaders you admire). If you have friends who are also quality developers, you'd ask them to do it. If they're not available, you'd ask who they know who might be good.
I've been consulting for 14 years. I had a tech website for a while, and though occasionally I got new clients through my website, most of my work has come through people I've worked with before, or people I know in real life, or people they know in real life, or know virtually.
Forget the freelance websites. As you've discovered, you'll never be able to compete on price. Instead, spend your time building relationships. If there's a specific industry you're wanting to target, get to know the people and companies in that industry. If you want to just work with any type of business in your local area, join local business groups. Network, network, network. If it's a specific technology you want to work with, get to know the leaders in that technology and your peers in the tech community. They may be able to throw work your way at some point, if they know, and know that what you're doing is quality.
Always remember, people are the ones who hire you, not websites, not even companies. These decisions are always made by a person, so focus on the people. Even in today's high-tech world, people usually choose to work with someone they know.