Tools like Adjust do not add noise. What they can do is make noise obvious that was not noticeable or barely noticeable prior to adjustment.
IOW, the noise was already there in your photograph.
Typically this happens when photos are under-exposed or lacking in saturation and then the photographer boosts the exposure/contrast/saturation significantly.
Sharpening can also take noise from barely visible to obvious, especially aggressive sharpening settings.
I have review copies of both Topaz Adjust and Topaz Vivacity. I believe you need to get more familiar with Topaz Adjust. I have not found it more likely than similar plug-ins to raise the visibility of noise. Some of the presets can be aggressive, but it's your choice to use them. In fact, I recommend manual settings rather than presets.