Can I assign this shipping date module to a shipping method?

Yes, to configure that please navigate to System > Configuration > Delivery Date > Delivery Date/Delivery Time/Comments Field and set to `Yes` the `Show For Selected Shipping Methods Only` field. Once you do that, there will be an opportunity to select needed methods in the `Shipping Methods` field:

Magento Delivery Date extension by Amasty

See more details on Delivery Date page
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