Yep, exhaust shutters...been there, done that. As stated, finding them where you did means at some point your engine was overheated.
They sit inside the vertical section of your exhaust after the downward elbow and pivot on a shaft. They are encased in rubber. The rubber melts and they fall down inside your exhaust. As you found, they can even work their way all the way into the stern drive.
They prevent water from backwashing into your engine which can cause hydrolock. This can break or bend things in extreme situations.
If you're cruising along and have to power down suddenly, a wave can hit the stern of your boat, flooding you exhaust system. The shutters come into play.
There are many different styles and sizes, so look closely at your old ones for comparison shopping. If you're lucky, there's a part number on them.