Bears, ha. They WILL freak you out until you get used to them. Then you realize they won't hurt you. Seriously, when was the last time you heard of a black bear taking on an adult human? Chowing on someone's leg? Those movies we all watched when we were little didn't do us any favors when it comes to bears.
A little kid alone may be a different story, especially in the spring when they are really hungry, after emerging from hibernation. The bears, not the kids. But they won't otherwise bother you any. I live with bears all around here, west of Concord NH, and I run into them on a fairly regular basis, once or twice a month anyway. My deer sightings usually last longer, because those bears-even the BIG ones-run away quickly. Like the one you saw did. They don't want anything to do with people, not the wild ones. The ones that get fed, that are used to people, they can make trouble, which is why no matter how tempting it is, you shouldn't ever offer a handout to a wild bear, period.
Once in a while, you'll hear about a bear following some hikers, but they just want the food they can smell (even though the hikers always say otherwise while changing their shorts), so its really about forest smarts and having some. If you're camping up over the top of Cranmore or other such place, you keep your food away from your tent, I mean 25 yards or more away, preferably hoisted up high about 15 or 20 feet overhead. Dangling from a limb, not against a tree trunk.
They won't bother you. You are more likely to need a woods gun because of a rabid animal, or maybe an aggressive moose, than you will need it for a bear here in NH. I carry a 4 or 6 inch 686 in the forests, and the only animal I ever drew on was a raccoon. I didn't shoot it. I've had a couple of other times my hand was on it, ready to pull it, but didn't have to. Not for any bear, which is the way I expect things to stay. Relax and enjoy them.
Btw, a 357 on a typical 250-300 pound bruin will be more than enough gun, if your loaded with the right bullets. HPs are not the answer, you want something that'll penetrate deep. Shot placement is key always. I carry 173 grain Keith's in front of 14.5 grains of 2400, in any L frame I own. Any Ruger will handle this load too, but I wouldn't use it routinely in any K frame Smith. or any Colt. My preference. The 357 is a great woods gun for this part of the world, it'll do anything you need it to do.
I can tell you, I am sure happy we don't have grizzly's around here though! Even that 629 would feel awfully small in 'griz country.