I can think of no better issue for the Congress to spend their time, for on the integrity of the family our society hinges, and we can trace many of our country's recent troubles on its weakening over the past half century. I agree that society hinges on the integrity of the family, but I wonder whether a Constitutional amendment will impact said integrity either way? Setting aside the fact that I don't think the state need sanction marriage at all (should be a function of the church), one cannot legislate morality and personal responsibility.
If your goal is to increase the level of personal responsibility within our society, I submit a more effective way to do so is to remove those mechanisms that allow IRresponsibility. Namely, the socialist programs that redistribute money and generally remove the consequences of one's actions. That's why I've repeatedly made the point on this thread that I see prescription drug coverage as far more important an issue than this proposed amendment.
I'd modify your proposal a bit. I DO think for legal purposes it makes a ton of sense to differentiate between mere "matings" and relationships from marriages and other binding commitments made by two people because our courts would simply be a mess if every broad I date asserts some claim :)
The simple way to do this is for a civil union to be enshrined in law, but not a license, merely a recognition of the marriage contract. Legally, it could only be between two people and would thus avoid the issue but not to the detriment of a smoothly functioning legal environment.