The basic shipping method I got is working at the cart page but it is not working at the checkout. Explaining further:
1. when the client is at the cart he can choose his location and the shipping methods and costs are working as needed
2. when he continues to the checkout and he changes his address town (or when he has not choose any shipping method at the cart page) the shipping method is not working
It like the CMS cant read the location change at the checkout page. I think its a bug or a conflict or something similar. It happened after adding some other plugins but disabling them does help any more.
If there was a change in address, it could be that the address is not supported by the Destinations specified in the shipping method Rates. If the address (state, country, postal) are the same then it would require investigation on your site.
You are right that it doesn't seem to detect a change, and update. I've found that deleting and re-entering the fields, recalculates the rates - but it is quite unprofessional to have to describe this to customers. And this is only if they ask - never know how many just thinks shipping is not available and leave.