Binary-continuous choice model in empirical consumer choices
It may not be a 100% match (and the details are a bit fuzzy in my memory), but I believe the model would be similar to Ericson and Pakes (1995).
In their model, there are $N$ firms who make sequential choices

Whether to enter the market or not (in your case, to consume or not), which is a discrete choice.
How much to invest if they have entered (in your case, how much to consume), which is continuous.

This is a seminal paper in dynamic discrete choice so I feel there should be some papers along this line.