If your grandmother has dementia,. she did not have the capacity to authorize you as a signatory on her card. Please consult with a probate or guardianship attorney.
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You have a tough situation since she has dementia. If she was not competent, the authorization may be ineffective. I would suggest that you sit down with your family and work out a way to repay the charges you made that is satisfactory to everyone. Certainly you were planning to repay those charges anyway.
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