I've not got experience with Goldsmith's, but I've had a tier 4 before and just sent off my application for my second one yesterday. If you're a US citizen, essentially the only thing you need to apply is the CAS. They reserve the right to request your documents for the qualifications used to get the CAS and for funding, but you don't have to send it in. So once you get your CAS, you do the online app, print it out, book your appointment, go to your appointment, then send your printed app, biometrics receipt, passport, passport photos, and priority receipt to NY. They say they process priority student visas in 3-5 business days, so you should be fine.
If you want to get ahead, you can go ahead and get your passport photos (I did mine online this time are passport photo.com but you can get them done at walgreens or those types of places). You can also go ahead and create the account on visa4uk and fill in the stuff you don't need the CAS info for. Like your personal details, travel history, etc. then it's just a waiting game for the CAS.
I'd stay on top of the school for the CAS if I were you. Though I would be too worried about time just yet.