For Nassau we take only essential items with us.
Our first cruise to Nassau we did the
DCL Atlantis Aquaventure port adventure. We rented a locker at Atlantis and put most of our stuff into it (this actually became a really good in the afternoon when there were torrential downpours... our stuff was dry even though we were not

) - I think the cost was around $10 or so and it was a good sized locker. At Atlantis and on the bus there and back we felt fine, but I did not like the brief walk outside the gates when we got back to the port. Fortunately it was really short. One nice thing with doing the excursion through DCL is that lunch was included, so we did not have to bring any $ for that.
Our second cruise to Nassau we just walked over to the Starbucks (which is very close to the docks) and then back to the ship. We were comfortable walking and it felt very different (in a positive way) than the first cruise we had done. We stopped and asked a couple police officers for directions at one point and they were very nice and helpful. After Starbucks we headed directly back to the boat, though there was a long long line to stand in inside the gates to actually board. The rest of the day we treated as a sea day and enjoyed the ship.
Our third cruise (part 2 of a B2B) we did the Blue Lagoon beach day, spending about 3 hours there before coming back. We enjoyed it. I didn't find lockers, so we just made sure once we found a couple really nice sheltered hammocks that we always had one of us stay with them -- it worked fine for the two of us. At lunch we just brought out bags with us, and we headed back to the ship soon after lunch anyway.
For future cruises we woulld just stay on the ship, or maybe take the walk to Starbucks. A note for Starbucks: you cannot use a USA/Canada Starbucks gift card there, nor the Starbucks App to pay. Credit card or cash.
SW