DH and I have done 11 Western cruises. We wander on our own every time.
Cozumel - we always walk to Mega (the target/
walmart of Mexico) first to pick up local craft beers, snacks and any necessities we might have forgotten. It's about a 2 block walk - just stick to the main road. Then after returning what we bought to the room we head back out again and do the main strip of shops. We might venture back off the main road near Palmeras where they have an open square with stores around it but we don't usually venture too more the 1-2 blocks back.
Grand Cayman to me personally is where I am most at ease. We have wandered several blocks into town and walked up and down the road every trip. A lot of people head out to beaches and excursions so the streets never seem that busy to me. They have a great brewery on the island, Caybrew, and DH and I hop a taxi there first to have some drinks and stock up on cans to bring home, then grab a taxi back to shop. They have public transportation right at the port as well and if you search it online, you can find their bus schedules and they are relatively inexpensive. I have thought about trying it to get to the brewery, but we usually just ask them to hail a taxi and pay the extra.
Jamaica's port in Falmouth is open and inviting. There is plenty of jewelry stores and souvenir shops to visit and the duty free shop. There is also a Margaritaville to hang out at. I have never gone past the gates unless it was on a Disney excursion. Last trip we noticed that they had security keeping the local people away from the port side of the street, where on other trips I had seen locals at the fence calling in to the tourists to try and entice them out. I have heard good and bad about going past the gates, but for me personally, there is nothing I need outside of the port itself so I don't bother. Friends on our last cruise grabbed a taxi to a local beach and had a fabulous time. I don't recall the name of the beach, but one of the crew on the ship was local to the area and told them exactly what to ask for and where to go.
One thing to keep in is be aware of your surroundings no matter where you go (when traveling and even at home) and pay attention to thing happening around you.
Have a great trip and enjoy exploring.