3S is any of the Lite-On 1213S/1613S/1633S/1653S/1673S/1693S drives, sharing the same chipset and hardware design.
Your drive may be fine, but it sure sounds like you are using questionable media.
The 2 letters(GG,TG,GD,etc.)+6 digits code is the sure way to identify real TY. If the discs are branded with That's, Fujifilm, TDK, or Sony logos, that provides additional assurance. ("Reusing" the TYG02 code may not be illegal, but putting other companies' logos on substandard media sure is!)
TDK discs can be either Taiyo Yuden (Japan) or CMC (Taiwan) manufactured. CMC disks will have completely different serial numbers beginning with letters on the transparent plastic near the center, so the lack of the TY code is not that surprising. CMC-made TDK are good-quality discs, too. Try burning them, see how they scan.
Test the drive with branded media such as Verbatim, Fujifilm, TDK, Sony. If they work well, you can then venture into lower-quality media, but I still wouldn't recommend it. Considering the cost of coasters and the aggravation of having to rewrite deteriorating discs a few months later or face losing the data, sticking with good-quality, branded media is well worth it.