Cambridge counseling? National debt relief... Will this negatively impact my credit?

Edit: I mean national foundation for credit counseling. Not national debt relief. I got connected with this company through them. It's a non profit.

We are in a lot of credit card debt and having a hard time getting approved for a personal loan to reconsolidate some. National debt referred us to Cambridge counseling who said they can put all of the credit card payment into one payment and negotiate much lower interest rates with the credit card companies but the accounts will close. She says it will show on our credit report that we paid them in full and not a charge off nor will any payments be stopped to the accounts. We will just lose them. We plan to keep a certain amount to not close and pay off with cash we have saved (but we are limited on this amount hence the need for consolidation). Has anyone done this route before and what did it do to your credit? We are considering this or paying off some debt and trying again to get a personal loan in a couple of months. We are just having a very hard time making payments monthly and don't want to get into a worse off situation by waiting.