Security Consideration on The Use of Random Number in Digital Signature
You should in general never use randomness from one cryptosystem in another cryptosystem. Bad things can happen.

Note that pseudo-randomness is good enough for cryptography, so generating new randomness is cheap.

Consider the specific example of using signed Diffie-Hellman with Schnorr signatures. Alice chooses a random number $r$, sends $g^r$, and later signs everything using by hashing $g^r$ and some other stuff to $e$ and then computing $z = r - ae$, where $a$ is her long-term secret signing key.

Now suppose the secret signing key is compromised at some point in the future. An adversary that paid attention recorded $g^r$ and the signature $(e,z)$. That adversary now knows $e$, $z$ and $a$, he recovers $r$ and can then recompute the shared Diffie-Hellman secret.

Just don't do it.