Does the server signs message body after ssl handshake?
..., is the message body (when my computer decrypts it) signed by server's public key or not?


It's not signed. The "message body" (i.e. application data) is only protected by a HMAC. This HMAC relies on a shared secret known both to client and server created together with the encryption key based on the pre-master secret generated during the key exchange. Given that client and server have both the same key for encryption and the sane secret for HMAC the client might create any encrypted application data itself. This means it is impossible for some third party to figure out if the application data in some packet capture originated from the server or was faked by the client.