You don’t need to turn off the “Driving” mode to make outgoing calls. And you can still access emails and messages as usual. The auto-reply will work as long as your phone in driving mode.
List the full name, phone number and email of an assistant or a coworker who can respond to messages and meet deadlines in your absence. If more than one person is handling your duties, list each one and the reason for contacting them. In some cases, you may consider providing a way to reach you in case of an emergency.

No one thinks much about cybersecurity while traveling. However, email security company Tessian warns the out of office message actually plays right into the hands of threat actors and cybercriminals. It’s a social engineering attack vector that no one thinks about. The out of office message is ubiquitous and handy. But if it includes any personal information at all — such as attending a funeral or going out of the country — attackers have all the information they need to impersonate the person who is out of the office, without the attacker having to do any real work.
Something that actually was an issue with my employer ages ago: at one time there was a policy that out of office would only go to internal people, and nothing would be sent at all to anyone external. Apparently this was felt necessary in order to mitigate risk of burglary, so people wouldn’t know that “John Winchester has gone on a hunting trip” and that his house was empty.