Hybrid or Road

If you want just one bike and you want that one bike to do be able to ride on the road when your riding by yourself, but also to ride off road when you're with your kids then you should look into either a hybrid or a cross bike which I think would be the better way to go then a hybrid. BUT, I noted you also want to do long distance touring, a cross bike is not suited for carrying lots of gear since most only have eyelets for rear panniers and no front provisions for panniers; also they usually have a higher seat to handlebar setting that is more road racing rather then a more upright position where the handlebars are level with the seat for more comfort touring. If you're going to be doing heavy touring then you need a touring bike. Touring bikes have similar tire widths found on cross bikes so the wider tire width helps with mild off road conditions.

The Fuji line is a good line. Do keep in mind that if you're planning on long distance touring you want to buy the best bike you can afford because you don't want cheap parts breaking on you while you're touring. However if the Fuji brand is what you want then I would take more of a serious look at the Fuji Touring bike not the Absolute due to your plans to tour.

Also take a look at the Jamis Aurora Elite and the Trek 520. Those three (including the Fuji) represent the best touring bikes for the money. Yes, you can spend more and get a better touring bike but you would have to spend at least another $1,000 to get anything significantly better. The Jamis comes with wider 32c tires instead of narrower 28's like the Fuji, the wider tire will make it easier to ride off road. Of course you should be able to put wider 32's on the Fuji but that will be the largest tire your going to be able to put on that rim, whereas with the Jamis you could go to a 37. If want a slightly narrower tire for road riding when your not touring you can put 25's on either the Trek or the Jamis.

I happen to like the Jamis the best followed by the Trek due to the Jamis more classic old school look. But the Trek has a sloping top tube which for some people it's easier to mount and dismount especially when loaded.

Once you decide on a bike you can do internet searches to see what upgrades most people did to their bikes to see if any of the upgrades is worth considering for your needs especially when you decide to tour. Also plenty of websites are available for helping you to decide which touring gear you should consider using.