Dodd is kinda iffy. The DLC stopped listing their membership and Dodd is on a list compiled on a MYDD blog
(LINK).
Edwards apparently was a member when in the senate. He gave a speech at the DLC's 2002 National Conversation
(LINK).
Excerpts from his speech
(LINK):
Responsibility of the kind we have seen in New York is at the heart of what the DLC has always stood for; it is written in the record and work of this organization. From national service to community policing to deficit reduction, the ideas you have advanced around the country have been about inspiring a new sense of responsibility in all walks of American life. Millions of Americans are able to lift themselves up, give something back, and hold their heads high because you have given them the chance.
...
And as we do more to strengthen our domestic defenses, we must tackle the ticking crime bomb in our midst: the 600,000 convicted criminals who are completing their sentences and returning to our streets every year. We have more than four million people on probation and parole who have broken the law before and are likely to do so again. We are putting criminals back on the streets and nobody's watching them. That is wrong and it has to be stopped. People on probation and parole who use drugs should be punished for it. We need more parole officers, out from behind their desks, and we need to hold people on probation and parole responsible for turning their lives around and becoming productive citizens again.
If we're serious about responsibility, we need to take responsibility for turning around our public schools, not walk away or leave them as they are. We need to ask more from teachers and pay them better in return. We need more choice and competition within the public schools, not vouchers that spend public money without public accountability. We need to do more to teach values at school, not less. The Ninth Circuit was wrong about the Pledge of Allegiance. This is one nation under God, and I want our schoolchildren to know it.