How much you choose to spend in port is up to you, as well as how you want to pay for it. Both ports gladly take US$s, and in fact prices at stores and kiosks frequented by tourist will have prices posted in US$. What you need to watch for is being given change in local currency, so try to have as close to the exact amount as possible. If you don't want to bring a fat wad of small bills upon boarding visit the casino and guest services to make change of large bills, just be aware some cruise lines limit the change to $10 in singles and $50 in fives per visit to the desk or cashier, so a few trips might be needed.
As for credit cards almost all larger "chain" stores will happily accept your card, but your card may want to charge you international transaction fees. Even when a business lists their prices in US$s their banking system communicates to your card issuer in terms of local monetary units. There are credit cards out there that waiver these extra fees like AmEx, Capital One, and other large financial institution issuers (but not on all their cards), so call to find out the fees structure. Never use a debit card, even a chipped one, as fraudulent charges do not have the same consumer protections as a true credit card. Also a good idea, even if not planning on using those accounts, is to notify all of your cards and banks of travel plans to avoid home based fraud while away.
A way to budget for ports is really no different then if you were staying a a hotel or resort on the island. Just research your prefered activities and experience expectations and what transport or formal tours to them will cost you. Then add in incidentals like snacks and beverages at a cost you would expect at a large amusement park. FOr us a general rule of thumb is to always have enough additional cash on hand while in port to be able to take a taxi back from wherever we have planned to go, and have 2 credit cards. One for large souvenir purchases (to us anything totaling $50 or more) and the second for emergency use. All other funds and cards stay in the in cabin safe.
Again how you wish to cover your vacation expenses is up to you. But do take some cash for on board spending, be it for extra tips or for the casino. One way to avoid having a large final bill while on the ship is to go through the various items offered in your Cruise Planner or Manage My Booking section of the cruise line's online check-in system. Unlike airlines the preplanning and check-in options start almost as soon as your have booked the cruise and close out 24-72 hours before departure. As part of this you can purchase things like beverage, internet, and photo packages, schedule and pay for speciality dining, excursions, and spa appointments, and have things like bottled water and souvenirs delivered directly to your cabin. Some cruise lines even allow you to prefund your general on board account, put money aside for specific uses like the arcade games, and a couple allow for wish list type of gifting that others can contribute holiday or other celebratory gifts/funds towards your cruise. The best parts are most of this will have some discount over the on board pricing and can be done piecemail buying the extras as you have the funding to do so.
Again how you wish to fund your on board account is fluid, and there is no reason to stick to just one funding method. Many experienced cruisers will do mix-n-match type of funding while on board, such as start with purchased or earned on board credits then a credit card on file as backup but before the end of the cruise pay the balance down in cash after all the ports are done. Some will put everything on a credit card that earns some sort of benefits for them, but have the cash or funds in their bank account to pay it off as soon as they get home. Our preference is to have a point earning credit card on file, that is separate from the card(s) we plan to take into ports, and pay off once at home.
Again avoid debit cards. The reason here is not about fraud but because the cruise lines (and I mean all cruise lines) do a test charge upon boarding to make sure a card is good. Then as you spend on board if you exceed the initial test charge more test transactions might be done. What happens is many banks keep these test charges on file as pending for 3-10 banking days, making it appear as if you have been doubled charged. This lowers the available balance interfering with your ability to use the actual funds in the account till the "holds" clear. A traditional credit card generally drops a pending transaction within 72 hours of receiving a finalized charge from the cruise line.