During the COVID-19 pandemic, people sought information through digital news platforms. To investigate how to design these platforms to support users' needs in a crisis, we conducted a two-week diary study with 22 participants across the United States. Participants' news-consumption experience followed two stages: in the \textbf{seeking} stage, participants increased their general consumption, motivated by three common informational needs -- specifically, to find, understand and verify relevant news pieces. Participants then moved to the \textbf{sustaining} stage, and coping with the news emotionally became as important as their informational needs. We elicited design ideas from participants and used these to distill six themes for creating digital news platforms that provide better informational and emotional support during a crisis. Thus, we contribute, first, a model of users' needs over time with respect to engaging with crisis news, and second, example design concepts for supporting users' needs in each of these stages.