Glad to hear you are cruising again. In ports where a tender is needed, the ship's tour groups get off first. Usually, everyone meets in the show lounge. Those who do not have a tour scheduled will get a number and they allow people to go down to the tenders according to the numbers. But, that is only for the first rush. After that, you just go down to the exit, usually the lowest passenger deck.
In Cozumel, and others, where there is a dock, you just walk off. But, there will usually be a line. Crowded at the start, then tapering off. If we are going off on our own, we always wait a bit to let those who are afraid to miss a single minute get off first.
Keep in mind that these two ports have become very popular and therefore can be somewhat crowded. Depending on which cruise line you are on, you may dock close to downtown in Cozumel or far enough away that you will need a taxi into town. In Grand Cayman, you are tendered right to the main town. Somewhat of a madhouse at times.
Either way, allow more time than you think you will need to get back to the ship. Traffic can be a problem, believe it or not.
Have a great cruise!!