Without knowing how the driveway came to be shared, it's difficult to give you any advice. Have you called animal control about the dogs being in your yard? You should speak to a local attorney for guidance, but just know that neighbor disputes can get very ugly. Your best bet may be to stop sharing the driveway.
The New Jersey Bar Association can give you a referral to a local attorney: www.njsba.com/
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.
Is there a shared easement over the driveway? If not, you may be able to apply for a dividing pony wall or some other means of dividing the driveway so they're dogs can't get into your fenced off area. Obviously this would limit auto access, so depending on the configuration of your yard/driveway, this may be practically unlikely even tho it might be legal.
If the document of ownership (deed) includes rights of sharing the driveway, they may be limited to a specific purpose, like pulling autos into the garage or parking cars...if so, you could fight them on THOSE grounds, too...and there MAY be grounds for a trespass action.
Given they refuse to be reasonable and won't move the line to where their dogs WON'T use your yard and driveway as their toilet, I agree with the previous commenter who indicated that you need to consult local counsel about your local rules...