I struggled with the idea of "our" money for ages, and it really only hit me when i went on maternity leave and had no income of my own..before then I'd just put half of all the bills, so for eg our rent was $220/week, so I'd put $110/week for rent into his bank account, and then it would all be paid out of his account. Now we have two joint savings accounts, we also have our own personal accounts/savings accounts, and his business bank account. He got extra bank cards, so we both have a card for his personal account and the business account.

If he doesn't like the idea of joint accounts, then duplicate bank cards probably won't work either..which would be much easier because you don't have to worry about transferring money into your account. Is there any reason why he's so against it? You two have been married for a while so I doubt you're going to suddenly run away with all his money!
If he won't agree, then you can at least work out how much your bills are per week and put that money into your account if yo uare the one that pays them, plus x amount for groceries, plus spending money for you (even if that's only $50/week) and that's your money to do what you like with, put into savings, have lunch with your friends etc but not for bills/groceries/essentials for the kids.