Is a digital signature of use if you don't also encrypt the message containing it? I'm sure this has been asked before. Its just too blindingly obvious.

from wikipedia

"Digital signatures, in which a message is signed with the sender's private key and can be verified by anyone who has access to the sender's public key."

So lets say Bob sends Alice an unencrypted message, and Charlie overhears it. What's to stop Charlie from signing his own messages with that private key and  henceforth having the ability to impersonate Bob?

So is there a point to digitally signing some message that isn't also encrypted? Or are all communications using digital signatures also encrypted? Wikipedia made it seem like there were two independent ideas, but I can't see a use case for signing-without-encrypting, where I can for encrypting-without-signing and encrypting-and-signing.
So lets say Bob sends Alice an unencrypted message, and Charlie overhears it. What's to stop Charlie from signing his own messages with that private key and henceforth having the ability to impersonate Bob?


Alice already possesses Bob's public key $\text{pub}_{B}$, and Bob is the only one that has access to the corresponding private key, $\text{priv}_{B}$. 

The private key is not shared with anyone, while the public key can be published.

Bob can use $\text{priv}_{B}$ to perform the sign procedure which creates a signature $s$ from a message $m$. 

Then Alice acquires $s, m$ and can use them with $\text{pub}_{B}$ to perform the verify procedure, which only outputs $1$ if $s$ was created using $m, \text{priv}_{B}$.

Charlie does not have $\text{priv}_{Bob}$, and so he cannot create a value $s$ for any $m$ that will pass the verify procedure that Alice will perform. (At least, he cannot do so with non-negligible probability, assuming that the signature scheme is secure).


  So is there a point to digitally signing some message that isn't also encrypted? 


Yes


  Or are all communications using digital signatures also encrypted? 


No 


  Wikipedia made it seem like there were two independent ideas


They are two independent ideas.

Digital signatures provide authenticity and integrity of the signed message - if the signature verifies correctly, then you know that the message came from Bob and was not tampered with.

Encryption provides confidentiality, which means that unauthorized parties cannot learn the contents of the message.


  but I can't see a use case for signing-without-encrypting, where I can for encrypting-without-signing and encrypting-and-signing.


There are plenty of use cases for signatures without encryption. 

Exchanging public keys for public-key encryption or key agreement is one of the most important examples (e.g. Public key Infrastructure).