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Jim VBThe answer to this comes down to your ability to manage your own finances and debt load.
As stated, the home equity loan is one option. A $20,000 loan, at 6% interest, for 15 years, costs about $170 a month to service. You can pay it off by stopping or reducing the purchase of new cards. But you have clearly stated that you are averse to incurring new debt and that's a sign that you know yourself pretty well.
However, as Johnny also stated, selling them as a group will never maximize your return.