You are in NYC? You are so lucky -- that's a hotbed of high-level dancing in the US. Try Ballroom on Fifth, or Dance Times Square, those two have some great teachers at them. There's others too, but I live in California so I don't know all the names of the best places. Maybe someone else here can help.
Competitions run between one and five days. Smaller amateur-only competitions run for as little as one day, usually on Saturdays, but larger competitions, inlcuding Pro/Am competitions (where a student dances with his or her teacher) run for several days and the American Smooth events are not always scheduled for Saturday. You can view a sampling of competitions and their schedules by going to www.accessdance.com or www.ndca.org