As I understand you are looking to rent, right? The Deerfield section of Short Hills has no apartment buildings. Renting a house will be the only option. However there are some apartments in the Glenwood, South Mountain and Wyoming sections.
All schools are great in SH/Millburn. Here is the Niche 2015 report with the Best Public Elementary Schools in NJ and all of our 5 primary schools were listed in the top 6!
Feel free to call or email me for more info or help finding a rental.