Isn’t ledger a cold wallet? It is definitely a cold wallet all hardware wallets are. Also avoid the misconception cold wallets still needs internet devices to help propagate their signed transactions to the network. No wallet does offline broadcasting of transactions. With ledger you get a device and install ledger live or wallets hot like electrum to how broadcast the transaction, the main hardware wallet which holds the private key only signs transactions offline and send it to the hot wallets either through usb or QR code this prevents the connecting of the hardware wallet with internet devices and there by keeping the private key off the internet which is the sole purpose of hardware wallets (keeping seed phrase or private keys stored away from the internet). So your cold wallet still needs internet devices for transactions broadcasting.
I wouldn’t blame OP much on the choice of ledger been is wallet, they were actually one of the sorted after wallets in the past before they screwed up and got exposed. So maybe OP bought it a long time ago. My advice is OP changes the wallet and go for another open source wallet.