The key is layers. You can pack long underwear [silk, for example] which will help in NY. Long-sleeved shirts, long pants, sweaters, fleece vests, etc., so that you can add/subtract layers as conditions warrant. For NY you will want a warm coat, a hat, gloves, preferably waterproof shoes. You don't say where in NY, but that may make a difference as to how much snow you'll encounter. Temps will be milder in Texas and Oklahoma, but won't be "warm".

Check the Weather Channel website or the NOAA website for info on predicted weather next week.

It will be VERY cold in NY. Shopping in NYC is going to be very expensive.

Yes I would buy winter clothes here over the Middle East.

Take it easy. Just bring cloths you are comfortable to wear. If the weather is cold there are lots of shopping center in where you can get one. New York is one of the best destination in US and sure you will enjoy shopping. Nice trip!