Get you some of them dog training collars for the kids and leave the shotgun out, unloaded of course. every time they even look at it, zap em. In fact, if even one goes for the gun, zap 'em all. You don't want to seem like you are playing favorites.
How one stores their gun(s) is a deeply personal and hotly contested matter. On one hand, barriers (cases, unloaded, locks) can burn precious time when glass is breaking at 0300. On the other hand, children are curious, and the more you hide something, the more they want to see and mess with it.
There are various ways to secure a HD firearm. You will hear a lot of elbows, they are like opinions, everyone's got 'em. Take it all in and decide for yourself what you are most comfortable with. I'd start with looking at long gun safes with biometric entry.