I'm not sure if your question is whether it is illegal for someone else to use your credit card, or if your husband could be criminally liable. First, it is definitely illegal for someone to use our credit cards without your permission. If it appears to be a problem with more than one card, you should contact the police , maybe they can figure out who is getting your information. As for the company claiming the calls came from your number, I have dealt with a case before where my client was wrongfully accused of extortion due to someone tapping into his phone line. If you have used the cards by phone, your card information may have been acquired that way, as well as the person using your phone line to make unauthorized purchases.
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You should file a complaint with the local police regarding these, I presume, unauthorized charges to your credit card. Also file a complaint with the Better Business Bureau regarding the issues with the 3rd company (one who is not refunding the charges to your account). You can also file a complain with the State PUC and Federal Communications Agency.
Ask this company you actually provide you proof of the location of the call, date, time and number called from.
Best of Luck