The Power Base Converter is just a slot converter that passes pins from SMS carts to your Genesis. There aren't any special circuits or anything, just some capacitors, traces, resistors, etc.

So, there may be a pin that your new games use that your old ones don't. The bad connection seems to be from your power base converter to your Genesis. Clean out your Genesis cart slot and the part of the Power Base Converter that connects to the Genesis, and see if that helps.