I've had that issue with dual signals in the past. The secret is additional compression on the clean signal.
The drive already has an element of compression on it as part of the overdrive. So the clean side needs compression so the note attack and decay react in the same way to the driven side.
It can be a bit of a sod to dial in, but when it is then it is much more like a single sound rather than 2. The clean also needs just a tiny bit of hair on it too. That sort of tone that sounds a bit dirty on it's own but clean in a mix like those famous isolated bass tracks.