Don't bury the lower portion. As you said, this will cause rot.

Thank you for a quick answer! I was thinking it may cause the lower portion to rot, but was hoping to somehow help the plant to not be as top heavy. Do you think the lower section will ever thicken up? The plant really doesn't have any problems keeping upright, it just looks funny. I have three of these plants, and one of them, having a similar appearance to the one pictured, bent over at 90 degree last summer. I thought it was rotted, so I cut off the top, where it was thicker, and rooted it. It is now growing very well. The surprising part is that the base plant didn't rot. It righted itself and has several new growth centers on it. I'm sure I could cut them and root them as well, if I so choose.

The base will not thicken, but I suppose you could cut and root it as you said. If you do that, just make sure you let it callous properly.

It is possible to use large stones to support the thin part and partially hide the thin growth without the danger of rotting. It also aids in keeping the plant straight until the roots establish.

Soil should be where it was prior to repotting.

The fatness up top likely is due to the growth receiving more light when the top part was forming. Think of the thinness as etiolation in your usual succulent plant.

If it bothers you, cut off the top and reroot.