This is a supplement to @Ken Graham's excellent in-depth answer. It's an expansion on a link he referenced in his second picture, which seems to perfectly fit the criteria you're seeking.
The organization is called Rail Explorers.
They provide guided tours you can take in a safe environment, with no investment or experience necessary. All equipment is included, and tailored to the number and ages of people using it.
The following information is taken from the various pages of their website.
About Rail Explorers:
Rail Explorers is a company that is passionate about sharing the fun and adventure of riding the rails. We provide a unique way of experiencing the outdoors in a safe, relaxed and enjoyable way.
The bikes, called Explorers, are different than true bicycles, and even true rail bikes, such as the bikes that serious riders like Ken Graham use. They have no handlebars, but they do have pedals and breaks.
What is a Rail Explorer?
A rail explorer is a pedal powered vehicle that rides on railroad tracks. They have 4 steel wheels, hydraulic disc brakes, pedals for each seat, and are comfortable, fun and easy to ride. Although the rail explorers require pedaling, the magic of steel wheels on steel rails makes the experience very different from riding a regular bicycle. There is no need to carefully watch the road ahead, there is no need to steer and riding is hands free - making it easy to take photographs and video as you roll along.
Choose either the Tandem Rail Explorer (2 seats) or Quad Rail Explorer (4 seats) for your ride. Single rider 'Jumpseats' are also available. (Pictures of the different models can be found here).
Who can ride the Rail Explorers?
Everyone! Rail explorers are suitable for all ages and abilities - the elderly, families, couples, cyclists, non-cyclists, hikers and couch potatoes.
As it says, it really is a true family experience. Children of all ages are welcome in the Explorers with enough seats so that they can be individually strapped in. Infants are allowed, and are strapped to adults using harnesses which are provided. You can even bring small dogs, if they're on a leash!
Each tour has a guide on their own bike in the front and in the back. They travel only on tracks without any trains, and have coordinated with local authorities to make sure no other vehicles are on the road during their hours of operation. Explorers are instructed to spread out, and are not allowed to pile up on one another, or "tailgate." That way you can feel somewhat on your own.
Tours run in all weather, as long as it's safe, and each bike comes with an umbrella! Trips are one-way, but they include a free shuttle bus back to the parking lot.
The company is committed to what they call "Green eco-tourism" meaning the bikes have no emissions; and because they use established trails, no trees, land or wildlife are disturbed to create the opportunity. This is obviously true of all Rail Biking, and another good reason to give it a try.
As @Charlie Brumbaugh found, Rail Explorers currently has a location in the Adirondacks Region of New York. Within the next few months, they'll also be opening another New York location, in the Catskill Mountains, and one in Rhode Island.
At the very least, this should be fun for your family vacation. Since your husband is already a bike rider, though, this might be just what he needs to encourage him to try altering one of his bikes, or purchasing something specifically designed for this purpose. Then he can get the "true" experience. Lists of local rail trails are easy to find on line, but I'm not sure if Rail Biking is allowed everywhere. There are authorities who can be contacted to learn that information.
Here are some fun pictures from people enjoying the ride. Click on them for bigger views.