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Of course you can take your own US Dollar & Euro's. Use your dollars in the Casino OR have Dollars charged to your sea pass account. Never change money on ANY ship or use the ship's ATM as they use a Gordon Brown type rate (a rip off).
The casino as the ship the preferred currency is dollars so you will not get the best exchange rate if you offer Euro's.
I am told that on Independence you can also use your sea pass card in the casino to get money to play and on the slots you get a printed slip(not coins) to take to the cashier or stick into another slot they are trying a Las Vegas method to cut down the cash being carried around,or you to loose more?