Does a digital signature ensure that the entire message is encrypted?
No. First of all, generally the message is first hashed using a cryptographically secure hash algorithm as part of the signature generation. The message itself is not converted to anything at all.
Even if the message is processed with the private key, anybody with the public key would be able to reverse the operation to get the value that is signed. And sometimes the verification just takes the hash as input and returns the result - as in true or false or an error or not.
As such, signature generation will never provide message confidentiality and therefore should not be considered encryption.