Well being an armchair lawyer...lol...I'll toss out what I see. I don't have the actual law to go by, but if the quote from the newspaper article is correct, everything above 7000 gvw RATING is banned in certain areas. It says nothing about what it is registered as, what it actually weighs empty or with just a driver, or what kind of cargo it can carry. Look at the data plate on the vehicle, if the GVW is over 7k, stay off that section of road.
NH has some foolish laws with regard to the actual data plate weight. For example, the trailer you are towing (weight RATING, not actual weight) can not exceed the trailer rating of the vehicle pulling it. So you can't pull an empty 2k lb. trailer that is rated at 7000 gvw behind a truck rated for a 5000 lb. trailer.
You also can't get out of CDL or other requirements by registering a vehicle for less weight. My friend went through this when he registered a 12,600 gvw rollback truck for 9999 lbs. He was stopped and ticketed for not having a medical card, fire extinguisher, flares, etc. which are required for 10k+ vehicles. He was registered non commercial, and just used the truck to haul his small tractor (loaded weight under 10k lbs.) When my friend claimed the registered weight, the DOT officer opened the door and pointed to the data plate and told him that is all they go by.
Laws can be useless, biased, and meaningless...but if you violate them, you don't have a leg to stand on. This is coming from someone who has been on both sides of many-a-law in my earlier days.
Jim