It sounds like the Rat might be more of what you are looking for. Blues can and does distort, but not THAT much. Try out a Rat, dial it back a little, then see what you think. Worst case scenario, you try stacking the TS and the Rat? Although, that may get untenable quickly.
I'll also put the Fulltone OCD overdrive onto your radar. It has a very wide range and my buddy loves it with his deluxe reverb. It isn't as classic or super well-known, but it isn't super obscure either.
I once met a guy in a guitar shop telling me about the "litter boxes" in the class room of a local high school. I was working as a substitute teacher at the time while I finished my master's thesis. I told him that I work at that school regularly and could confirm that it was a facebook shitpost and not even remotely grounded in reality.
He walked out of that store still 100% convinced that our local high school just had students casually pooping in the corner during class.
I mean sure, we still had students pooping in the corner during class, but it had nothing to do with litter boxes. It was very much motivated by how much they hated that particular science teacher, and there was absolutely nothing casual about it.