Is there a way to separate geometry that shares verts/edges? I thought it would be cool to use random selection and extrude along normals to make a sci-fi handle grip. Later, I found out that it didn't work quite the way I hoped and cannot be beveled. Is there some way to separate vertices like this:

Here's a quick example of the end result I hope to achieve.

I know I can extrude along individual faces instead of along normals, but I still want adjacent boxes to be attached.
There are too many to bother doing this by hand:

I'm still pretty new to Blender, so I appreciate any and all help/advice. Cheers :)
Once gone into the Non-Manifold mesh path, it's hard to get back.
I suggest an hack: after the selection, Inset faces a little bit.

Make sure that Individual checkbox isn't active:

Now, you have disconnected geometry you can easily extrude and bevel.