Bartimaeus is onto something. We're all assuming (and I mentioned before) that the distortion you speak of is due to overwhelming a stage with too much signal, resulting in clipping due to lack of headroom...exceeding power supply's ability to amplify due to this limitation (asking for 12v when only near 9V can be supplied). I mentioned in a prior reply - there is overdrive/distortion occurring normally in the clipping stage (hence its name). You may not be aware of this (that 'assumed knowlege' thing), but how you shape the tone will make that distortion more or less apparent down the line.
If you cut highs after distortion, it decreases the apparent level of that distortion; it makes it 'wooly' and less biting. Adjusting mids changes its character quite a bit. And so on. Low end has the most energy, so overdrives following stuff more, requiring more from the power supply. So, when you increase the mids above a certain point, and you 'get distortion', I am assuming this is due to overdriving its associated opamp. This is exactly why I asked you to listen to each stage with an audio probe to determine what is 'normal', and what is being caused by your adjustments. I don't know if you did that or not (and should be doing it at every step here), but that's our working assumption.
Not an easy task to undertake, is it?

My preferred route for this is 'dirt box > EQ pedal' and done. I rarely bother with anything more than a high cut to knock back some shrill. Or, guitar>EQ>interface>DAW>sim>EQ...
TS is a one-trick pony, and there isn't much special about it. Most any feedback clipper followed by some form of EQ circuit would get the job done, IMO (ROG's "Tone Mender" comes to mind). But this is (hopefully) a learning experience.