I wouldn't call myself an expert, but I have coded for urgent care and my experience has been that the coding can be largely driven by how your payer contracts are set up. My suggestion would be that S9083 only be used if that code is specified in your payer's contract or provider billing manual as how they wish an urgent care visit to be billed. The S code is used by certain payers that contract for a specific flat-fee case rate for an urgent care visit. This would replace the E&M code that your provider would otherwise bill. For Medicare and payers that don't direct you to a particular way to bill for urgent care, your coding will be essentially the same as any other physician or facility coding.
You're calling your urgent care center a 'facility' but that can have different meanings. Whether or not it is owned and operated by a hospital, and if it is on or off campus, will be a big factor in determining how your claims will be billed due to provider based billing rules for facilities. If it is not a hospital-owned facility, the coding and billing should not be much different from billing a private physician practice. Hope this helps a little.