Might want to consider that some of what a typical shotgun class may teach doesn't necessarily apply to most homes.
Transitioning from shotgun to handgun would be one of those.
I can't see much use for a sling on an interior home defense shotgun. In most homes, distances are short and spaces (hallways, rooms, etc.) relatively cramped. If I were carrying a shotgun & a handgun in my home, and further action was needed after emptying the 12-gauge, it'd be immediately dumped on the floor and the handgun would be up and running.
I would never bother to take the time or effort needed to sling it or hang it in front of my torso where it'd be an empty dead weight while trying to move as needed.
I'd say most classes that teach transitions intend them for more open spaces. Nice to know how, but in my 1600-foot upstairs & 1600-foot downstairs home, not practical.
Those classes are also typically designed for more LE & military applications. In such environments, engagements may or may not be more prolonged, and dropping a weapon (even if empty) would not be encouraged for various reasons.
You have to get what works for you out of a class, and apply or modify it to your own abilities & needs.
A sling inside your own home has very little real utility or benefit. It can snag on items as you move, and it provides another grabbing point for anybody who can get within reach. There are a couple of simple & effective disengagement moves to get a shotgun out of the grasp of somebody who does manage to get their hands on it (usually they'll grab the barrel), but it's a little harder to disengage if he grabs your sling.
A slung shotgun can also be a very effective come-along if somebody gets within wrestling distances. Remember- the gun's now attached to YOU, and if anybody grabs it they can toss you all over with it.
In your own home, there's little reason not to just dump an empty shotgun in favor of the handgun, if you have both with you. You won't be moving far, leaving the area, abandoning company property, needing it for the next engagement next doors, providing a usable weapon that can be immediately turned on you or family members, and so on. The exception, of course, would be if you had additional rounds in something like a Side Saddle that could possibly enable an adversary to pick up the gun, load it, and use it.
If you evaluate your own environment honestly, without allowing the "experts" general principles to override common sense in meeting your own specific needs & abilities, and you feel a sling would offer some advantage to you, by all means go right ahead.
I've taken classes, I've taught classes, and what I do in my own home doesn't always match up with those.
Denis