Attack on DSA modification with bad hash function
With my new idea I seem to solve the problem and answer my question, so I'll go ahead and post it as the answer. I choose the new $m'$ as $m'=t+m$ with $t&gt;0$.
Now the verification works like this:

$v'=g^{m'w} y^{rw}=g^{m'w+xrw}=g^{tw+mw+xrw}=g^{tw} \cdot g^k=r' \mod q$

So my new $r'=g^{tw}r=g^{ts^{-1}}r$

This means I can create a legit signature to any $m'=t+m$ which is $(r',s')=(g^{ts^{-1}}r, s)$. Correct?