Is it wise to do a money exchange or just use US$.? I realize that our money is not worth as much in Cayman.

The CI$/KYD has a fixed cash exchange rate with the USD. No need or benefit to exchange money if you already have USD.

$1.00 KYD = $1.25 USD
$1.00 USD = $.80 KYD

Note, when paying with USD you will get any change back in KYD.

Credit cards are also almost universally accepted (Visa & Mastercard, Amex less so) and transactions for US based card are conducted in USD. So if you have a card that does not charge a Foreign Transaction action fee it can be an easy alternative to carrying cash.

.... and, unlike purchasing with a credit card in other countries; there is no better exchange rate provided through the credit card company's rate. U.S. cards used in GC retain the flat 25% rate. On my trip last Feb, rarely if ever used cash, used credit card; I don't even recall seeing GC currency or change.