I live on St. Croix. Typically, except for peak season in the winter, you can play it by ear and wait in regards to reservations, diving, and excursions. However, we've been experiencing a an EXTREMELY busy offseason in tourism. Restaurants that we normally could walk up to and get a table months ago now are booked several days out. Casual dining places (e.g. Shupes, the Mill for Pizza, Brew STX) are still available to walkup, but more formal places like Hamiltons, Savant, and Duggans will require advance reservations.
I've also seen popular excursions such as Buck Island have been booked out up to a week or more in advance leaving people who waited to get here to make reservations disappointed. Same with rental cars being difficult to find without advance reservations. For the rum tours, the Cruzan distillery is still closed to tours I believe, but have heard that Capt. Morgan reopened, but only with advance reservations and it has limited availability. However, our local vodka distillery and brewery - Mutiny Vodka and Leatherback Brewery - are both great spots and don't require reservations.
If there are things you really want to do here, I'd make reservations in advance. I'd also check with the dive shops you're using to see if their schedules are filling up and at least reserve some of your dives. You don't need to plan out and make reservations for every moment of your trip, but I'd plan out and make reservations for your "can't miss" places. Then there are plenty of other things you can do that don't require reservations or planning to fill in the gaps in your schedule (e.g. visit several of the many beaches around the island, day trips to Frederiksted or Christiansted to visit the shops and historical buildings, hikes, etc.).
Enjoy your trip and let us know if you have any other questions.