Once the immediate danger has passed, the situation stops being an emergency and becomes a problem to be managed. The habits change.
Set a rhythm rather than watching continuously. Check official sources at fixed intervals — every hour, then every few hours as things settle. Continuous watching exhausts you, drains the battery and does not make information arrive sooner.
Ration the battery from the beginning, not from when it is low. Aeroplane mode between checks, screen dim, one device on and the others off.
Have a radio that does not need the internet. Mobile networks and home internet fail together in almost every large incident. Broadcast radio keeps working, and several national civil protection agencies specifically recommend a battery or wind-up receiver for this reason.
Plan in twelve-hour blocks. Where will we be tonight? What do we need before then? Who needs to hear from us? Anything further ahead than that will change before you get there.
And write things down. Under stress people forget what they were told twenty minutes ago. A note of what you heard, from whom and when is worth more than you expect on the second day.