I’m not convinced that New Yorkers smell worse (or better) than the average American. Yes, central A/C in uncommon in residential buildings, but that’s pretty normal for the Northeastern US. It’s not really needed in winter, since if you take 35 degree air at 100% rh and heat it to 70F, you end up with about 25% humidity (aka too dry). You’ll get some water added from showering, cooking, perspiration, exhalation, but it will still be very dry inside — if anything, you’ll be more likely to need to stick a pot of water on a radiator to add humidity, not subtract.
Discount stores. Hmmm, IKEA in Brooklyn, H&M in several locations in Manhattan, there’s a Kohls in Queens, Target on 115th(?) St in Manhattan, Costco uptown as well, bunch of local, inexpensive hardware/houseware stores. Walmart is MIA, but that exists too in the NJ suburbs.