How are you going to sign a message if you're not willing to use a website or some software?
Well, okay, I give it to a software, namely one on the airgapped system.
Maybe this is the part you're not understanding. What do you mean about "giving up my sig address"? Do you think that when you sign a message with an address you aren't making the address known? Because you are. Every time you publish a signature you are publishing the public key along with it.
I was referring to the vanity address in my signature. If I'd use some website which claims clams by using my private key, that address would essentially be dead. And I want to actually keep it because it's well published.
Signed proof is how every transaction works in every crypto. If you're looking for a way of signing a message without revealing the address used to do the signing I don't think you're going to find it.
With Electrum, you can have two wallets, one with the private keys on an airgapped machine, and one with watch-only addresses on a connected one that does not contain any private key.
You can create a raw transaction which you can transfer to your secure machine, sign it there, copy the signed transaction back and broadcast it. That way, the private key was never exposed to the potentially insecure machine.
You can even copy that signed tx to a pushtx website, like at blockchain.info.