I run VCAS slings on all my AR’s and shotguns. On the 870 that has the Magpul SGA, I also have this front sling mount: http://www.gggaz.com/remington-870-quick-detach-front-and-rear-sling-attachments.html
Be advised, if you use that front mount and want to use a Surefire 618 light, you will need to grind off part of the mount. Just 10 minutes with a Dremel - pretty easy.
On the rear, I have the sling looped through the slot on the right rear of the stock. It works fine for me, and does not have any tilting issues as you describe them. YMMV.
On the other 870P I have, I run a Vickers padded Cobra sling: http://www.blueforcegear.com/vickers-padded-cobra-sling.html
Up front I use a Universal Wire Loop like this: http://www.blueforcegear.com/universal-wire-loop-uwl-3-25.html
And on the stock I use a BFG A2 stock adapter like this one: http://www.blueforcegear.com/a2-buttstock-adapter.html
And I use the HK style hook to connect to it: http://www.blueforcegear.com/hk-style-hook.html
These give me the ability to ditch the long gun if I need to, but also makes for a very adjustable way to carry it. It hangs very comfortably, and I can easily operate the controls or reload from that position.
I have a friend that is a fan of a single point sling on his SBS to hunt bad people inside buildings. He’s done that for a lot of years, and it works for him. I’m using the shotgun from a fixed position barricaded inside a master bedroom, so I’m just looking for a sling the way you would use a holster. Different needs, different tools. As an LEO you need to go looking for bad people, and I do not.
And no, the shotgun is not my first or second choice, for all the folks about to chime in. But these particular slings and hardware have worked well for me in classes, and manipulation drills at home, plus testing on the range.
I have a third social shotgun, but it does not wear a sling at all. It’s a Winchester Model 12 that breaks down for very discreet carry when traveling.
And it may just be me, but I am not a fan of the Spec-Ops slings. I tried one of their two-points, and was just not impressed. BFG with their last call page is a great way to pick up an excellent sling on close-out. They are first quality, just in a color that is no longer being stocked. I could not give an obese rodent’s bottom about color. I bought of ton of their stuff when they phased out OD green.
Here’s some for $30! http://www.blueforcegear.com/last-call Very sweet.
Hope that helps, and stay safe.