Part of the confusion comes from the fact that the same statute covers possession of both HP and AP ammo. The statute is 2C:39-3f, and is titled, "Dum-dum or body armor penetrating bullets". Paragraph (1) applies to HP ammo, while paragraph (2) applies to AP. The AP restriction is pretty much a mirror of the federal AP restriction (meaning, handgun ammo).
Since the woman was in possession of a handgun, and did not possess a NJ Permit to Carry a Handgun, and was not operating under one of the narrow exemptions provided for in either 2C:39-6e or 2C:39-6f, she was charged with violation of 2C:39-5b, Unlawful Possession of a Handgun.
Since her handgun contained hollowpoint ammunition, and she was not operating under one of the exemptions provided for in 2C:39-3g, or 2C:39-6f, she was charged with 2C:39-3f.
No, the fact that she possessed them in her car has no bearing on the charge, as NJ does not consider your vehicle to be a "dwelling".
And, even if she wanted ot come up with some after-the-fact explanation of going to a range (which would fall under the 2C:39-6f exemptions), she'd still have been charged, as her transport did not conform with 2C:39-6g (firearm unloaded, and contained in a securely fastened case, or locked in the trunk). The firearm was loaded, and contained in her purse (which she was rooting around in to retrieve her lic/reg/ins).
Realistically, her only hope is that the media attention pressures the county to re-evaluate, and allow her into the PTI program. Otherwise, she will be found guilty, and will be sentenced to a prison term of between 5 and 10 years, while having to serve a minimum of 3 years, due to another law (the Graves Act, 2C:43-6c).
Brian Aitken was a very similar case to this, and he wound up getting sentenced to 7 years (serving concurrent sentences for the unlawful possession of a handgun, unlawful possession of a large capacity magazine, and unlawful possession of hollowpoint ammunition).
Noteworthy is the fact that, after Aitken had his sentence commuted (not pardoned) by Gov. Christie, upon appeal, his conviction for the handgun was overturned due to his defense not being allowed to introduce evidence that he was moving at the time (which is a permitted exemption under the law). His conviction on the magazine was overturned, and the charge dismissed, due to the fact that the prosecution never proved that the magazine was capable of feeding more than 15 rounds directly into a semi-auto firearm (the arresting officer verified that the magazine held 16 rounds, but it was never test-fired). The conviction on the HP ammo remained, because even if it was concluded that he was, in fact, moving - there is no exemption to the law prohibiting HP ammo possession for moving from one residence to another. While it's legal to transport your handgun, you can't transport your HP ammo.
All brought to you by one of the original 13 adopters of the United States Constitution, and her included Bill of Rights.