Hello, I'm pretty new to this board, but I'm glad I found it because of the wealth of knowledge here. I don't have much experience with TG16 systems, but I have modded several Sega Genesis systems, as well as an NES and two PlayStations. If I were in your shoes here's what I would try.
The TG16 motherboard has 5 or 6 surface mount ICs on it. I would use a fine tip soldering iron to carefully touch and reflow the solder on each pin of all the surface mount ICs. It would seem to me that it's possible there are one or more cracked solder joints that could be causing this.
Of course, what I'm recommending may be beyond what you're able or comfortable to do. First it requires a special bit to open the system up. You'll also need a good solding iron and wick to be able to remove the RF shield. And you need a small tip for your soldering iron so you can reflow each pin without bridging to the next. But assuming you have all this, you might give it a try.
Again, I'm by no means an expert when it comes to repairing these systems, this is just what I'd try if I were in your shoes. Good luck!