How does 5 days suit you? If you bypass the appraisal, most attorney's can have a cash deal done an under a week. I would also closely examine the builder's language for their 1 year punchlist. Typically, I don't recommend buyers of new construction homes spend the money to have a pre-closing inspection (pre-drywall of course) since everything can be inspected by the homeowner prior to moving in. However, before you have your 1 year punchout, hire the inspector to see what needs to be addressed at that time.
Inventory homes can be a great deal and Fort Mill has its share. It's really a case-by-case basis with each builder however you should be able to negotiate a win-win for both parties. The more a builder has in inventory, the greater the likelihood to negotiate. Clearly though, with such a time crunch, the best thing would be to have a list of places, previewed by an agent with all the details beforehand so you don't waste your time on "bad" properties.
I'm licensed in both states and a member of the local association that serves York and Lancaster County (Fort Mill) and have access to ALL the listings (most agents only have access to the Charlotte area MLS, which isn't the main MLS for Fort Mill and York County). If I can assist you in any way, don't hesitate to ask.
FYI: Be aware that the property taxes on your new home will be based directly on the price that you paid for it at closing. Its a new law for South Carolina that most people are unaware of.
Finally: Forgot to add if you do opt for an appraisal (which is not required for cash but highly recommended since any home, new or existing can fail to bring the appraised value), you can still close within 7-10 days. Even though Fort Mill is "hot" for this area, we're still in a "buyer's market" with the average home in that area taking over 130 days before a contract on average.