I no of no rule that prevents someone from applying to a professional school if they have a drug conviction. The issue arrises from the licensing board in the state that you want to practice. If that board says that no one with a ______ conviction, even if it is expunged, may apply for licensing, then you will have a problem. Since the school doesn't know where you will apply for licensure, it is unreasonable for the school to use licensre as an issue to deny admission (at least openly)
Most schools rely on questions in the application and, perhaps, a quick internet search, to see if someone is not ethically pure enough for admission.
Licensing Boards almost always require fingerprints and a sworn statement when you apply. Some also do internet searches or pay for on-line data searches.
Employers do everything from nothing to hiring a private investigator to probe your past. If you work for the government or a government contractor, expect at least a fingerprint check.
My advice would be to pay for an online search of your name and see what shows up. If the conviction appears you should contact the Court about having the conviction expunged or the charge reduced. A lawyer may help you do this.
I would definitely recommend looking into expungement proceedings. Depending on the nature of the drug conviction and how much time has elapsed since your conviction, you may be a good candidate for an expungement. A competent attorney in New Jersey who regularly deals with these issues should be able to tell you over the phone whether you (1) qualify now; (2) don't qualify at this time; or (3) whether you will or will not qualify at any time.