I'm due March 3rd and everything I have bought in size newborn is long sleeved or pants; anything I have purchased in 3 months is short sleeved or shorts. More than likely he will be out of the newborn clothes by two months at the latest so I don't think we'll run into too many issues. I'm in the Dallas area and in March it can range but realistically we won't leave the house until April with the exception of a few doctors appointments, and even if we do I can throw a blanket on him if it's still chilly, or leave it off if it starts getting warmer.
You can always buy short sleeve onesies and some zip-up hoodies/cardigans, they have some (hoodies) at Carter's right now for like, $8 I think- and that way you can still use the onesies into the summer (if they still fit).
DD was born in June and it was already a sweatbox outside, she would be in a sleeveless one piece and sweating in her carrier so you never know-