Are there functions in the same Θ-class that are not linear transformations of each other?
Your proof must be wrong since the result is incorrect. For example, if $f=x^2$ and $g=x^2+x$ then $f = \Theta(g)$ but $f$ is not an affine shift of $g$. To see where the proof goes wrong, I suggest substituting these $f$ and $g$ in the proof and seeing which step fails. This is how one debugs their proof.