I read that the same key should not be used for both signing and encrypting.

I wonder if this may not be a problem with PKI/TLS/SSL because asymmetric encryption is used to transmit the public key, which is used to encrypt a browser generated session key, which is then used for symmetric encryption of the actual traffic?

Is that how it works and does that mean it is okay to sign both with the same key?

How would I go about using separate keys?