Credit cards are only useable in large hotels and ATM's and it is not unusual for these to have downtime where they are out of order. Find an ATM in the airport on arrival and withdraw TZS, then pay your taxis in cash TZS. While most places on Zanzibar will happily take your USD cash, they will absolutely rip you off with exchange rate and inflated prices. There is currently a shortage of foreign currency in Tanzania and while the official exchange rate is around 2,600 the unofficial rate is pushing 2800-2850 as we speak.
Last time I did the trip from Stone Town airport to Nungwi I think I paid around 60-80,000 TZS, that was a few years ago though. But since I assume your Kiswahili skills are not as good as mine, you will likely end above this price. TZS 80-100,000 should be fair.
The drive is around 1:45 hrs. Ask them to drive slow and careful, accidents are very common. Also avoid driving long distances at night, for the same reason.
Your hotel should also be able to arrange transport.