When requesting shipping rates from ShipperHQ the rates will be calculated based on the origins, carriers, carrier rules, boxes, etc for that exact cart and shipping address. The resulting shipping methods and respective rates are then stored in a cache server-side for a short time. This is by design and cannot be disabled.
What this means is that if a customer requests a shipping rate multiple times for the same cart and shipping address the shipping methods and rates will not only be displayed much faster but also this would not count against your monthly request limit.
The caveat of this is that during testing, where shipping rules are being modified, boxes are added, etc, you will see cached shipping rates when re-testing as the cart contents and shipping address, commonly, will be the same. To avoid the cache and receive fresh rates just change any aspect of the cart or address. This could be the quantity of products in the cart, additional/removed products or something slight, such as changing a single character of the ZIP/postal code.