Does a Bitcoin hardware wallet need the internet to sign a transaction?

The answer

No. An offline signing device can authorise a transaction without an internet connection; a connected device is then needed to broadcast it.

What to do next

Use a documented PSBT or QR-signing workflow, inspect outputs on the signing device and broadcast only the signed transaction.

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Can a watch-only Bitcoin wallet spend funds?

Not without access to the required signing keys or a co-signing device.