Thanks for the suggestion, Akshat, however, your suggestion will not satisfy the use case. Here, the same work queue needs to be used and there's the need of assigning the sticky cases as well as the new cases when the agents are in the "Available" status, BAU, but when they are in the alternate status, they should receive only sticky cases. The motive is not to split the new and sticky cases into two different work queues and switch between statuses to take up only the new cases while on one availability status and only the sticky cases while on another availability status. 

We are going to use a workaround based on some available rule conditions derived from the "Assign Fairly" approach, to directly assign the cases to the associates wherever applicable based on stickiness and the alternate status. I'm going to test the build and share more details if it works out.

Assignment happens to a agent only when the agent is in one of the statuses in "Agent status when available". So clone the queue, tag the sticky cases with a custom field and put them in this cloned queue.Keep the status you want in the "Agent status when available". Otherwise, the cases will go to the holding queue and be assigned when agents become available.
They would need queue control as well, to stop getting cases from the main queue.