As of late, I have mailed bullets that I have sold on Gunbroker.com to different states several/many times. Bullets are inert and are not a hazard to the mail carrier. The most cost effective is a Small Flat Rate Priority Box. Being flat rate, the weight does not effect the price ("if it fits, it ships"), inside the USA. The Small Flat Rate box costs $5.80, available at all Post Offices, or $5.15 if the label is printed out at home on your printer. You can down load the program to do the printing and charging your credit card at home.
As to the legality (State law of New York), of shipping bullets, I do not know. But sense there is no requirement to label the contents of the package as "bullets" (or anything else), the State of New York would not know what was in the package...even if they had access to it.