Hey BELZAN,
Great details. If we understand correctly, it is no longer auto-dialing emergency services, but your emergency contacts are being texted. This also seems to happen in the same spot, while traveling. Is that correct? If so, if you haven't already, restart your iPhone to see if that helps with fresh connections, etc.: Restart your iPhone
Then, it's also a good idea to test without your case and with a different USB cable. This can help rule out both as being the possible culprit. An update was also recently released. The current version is 13.6.1. If that isn't what you show in Settings > General > About, we encourage you to back up your iPhone, then update the software. Just in case, here's how: Update your iPhone, iPad, or iPod touch
Let us know how that goes.
Cheers!