Post a high resolution version of your schematic. The exact blog post where I believe you got the pixelated one from has a higher resolution version too.
W.r.t. your question, as the circuit for buffering the sawtooth is just a voltage follower connected opamp, any distortion will be dependent on the specific chip used. Are you using a genuine TL07x?
The same goes for DC offset on square. The circuit shown is just an opamp buffering. If it must swing close to the rails, how close it can get depends on the amplifier specs. In the schematic I found, this was also a TL07x which is not able to swing closer than 2 or so V from either rail. Again, are you using genuine components?