Emerging experimental evidence shows that writing with AI assistance can change both the views people express in writing and the opinions they hold afterwards. Yet, we lack substantive understanding of procedural and behavioral changes in co-writing with AI that underlie the observed opinion-shaping power of AI writing tools. We conducted a mixed-methods study, combining retrospective interviews with 19 participants about their AI co-writing experience with a quantitative analysis tracing engagement with ideas and opinions in 1{,}291 AI co-writing sessions. Our analysis shows that engaging with the AI's suggestions -- reading them and deciding whether to accept them -- becomes a central activity in the writing process, taking away from more traditional processes of ideation and language generation. As writers often do not complete their own ideation before engaging with suggestions, the suggested ideas and opinions seeded directions that writers then elaborated on. At the same time, writers did not notice the AI's influence and felt in full control of their writing, as they -- in principle -- could always edit the final text. We term this shift \textit{Reactive Writing}: an evaluation-first, suggestion-led writing practice that departs substantially from conventional composing in the presence of AI assistance and is highly vulnerable to AI-induced biases and opinion shifts.