Our Costco card is under my wife's name. Do I need to pay for gas with her debt card or can I use mine even though I'm using her membership card?

Tom is incorrect. You slide her Costco card in and then slide a payment card. Doesn't matter whose card it is. Do yourself a favor and get her to let you on the account. It is worth it. It's free to add a spouse.

Name on card has to be the same as the credit/debit...as I have seen it.