I'm afraid you might have to twiddle quite a bit with this.

As a first step, you might try playing with friction coefficients, but it would clearly have other unintended consequences.

For 2D graphics, there's a way to disallow 3D rotations, allowing the objects to rotate only around the "in-out" axis. Maybe you could try this as well, but I don't think it's going to work.

Eventually consider a more generic system to provide the extra "stickness", such as adding and removing constraints on the fly. Sounds more robust to me but sure it's still going to take some care.

I'm vary interested however so I'm looking at this thread.

Moving dynamic objects is discouraged (you'll wreck arbitrary internal state). At best, inject forces (even this will eventually give issues). So I'd say that's a recipe for trouble.