What you are doing is exporting your key. Your public key is indeed something you can (and should) share as it enables others to verify that you are indeed who you claim to be (or more accurately, that you’re in control of the private key that’s linked to that public key). So while you should share your public key, your private key must remain private.
What these people on the dark web are doing is one step further: they sign their messages with their private key. This creates a cryptographic signature that’s different for each message (changing a single character in the message will generate a wildly different signature). Anyone with the public key can simply copy that message including the signature and validate it. If even a single character of the message was changed, the signature will not be valid. Thus ensuring others that the person who posted the message is indeed in control of the private key.
Signing is different from encrypting: while encryption renders your message totally unreadable to anyone without the correct key, signing doesn’t change the message itself. It simply appends a signature allowing others to check that the message wasn’t tampered with.