In response to your question, whether you can have different due dates for different checklists ... no, you can't, at least not with Trello's standard features.
But there be a workaround: You could use custom fields and set up one or several date custom fields that capture due dates for the checklists you have. Custom fields can be displayed at the front of the card, so the date would be somewhat obvious.
When using the Butler Power-Up, you could send yourself (or card members) notifications for due dates, or similar. So I could imagine this task can be tackled somehow.
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