If I understand correctly, to claim CLAMS I only need to provide a signed message from the respective BTC or DOGE address. Is that correct?
Nearly, but not quite. You need to provide a signature, but of a transaction, not a message. This is the same that Bitcoin uses - when you want to spend your Bitcoins your wallet makes a transaction and signs it using your Bitcoin private key.
The point you may be missing is that the private key for your free CLAMs *is* your BTC private key. The private keys are the same, and the addresses only look different because of the 'version byte' (the thing which makes BTC addresses start with a '1' while CLAM addresses start with an 'x').
So the easiest way to sign a transaction using your private key is to import your old BTC/DOGE private keys into a CLAM wallet, let it rescan, and then send a transaction from the CLAM wallet as you would from any other wallet.
If so, could you please tell me the command to use with the server daemon for doing this?
"clamd sendtoaddress
" - or use the QT version and use the send tab.