Does anyone know how to increase the thickness of a shape (not a stroke)

NB,

Strangely, the shapes in your attachments are fundamentally different to those shown directly in your post:

In your post, the top left and bottom right corners are (almost) sharp, and the end corners are sharp, and the top and bottom horizontal parts are thinner than the parts in between.

In contrast in the two attachment shapes, all corners but two are rounded, and the all the parts have the same thickness.

So judging from your post shapes, you seem to wish to keep the sharp corners and thinness of the top and bottom parts, and increase the thickness of the parts in between, with the corresponding change of the overall shape which is obviously wider.

Obviously, this requires a customized solution.

The shapes in your attachments are fundamentally different, suitable for a simpler solution.

At first glance my impression of your post shapes was that you wished something somewhat like the second image, only more consistent including identical thinness of the top and bottom parts.

​@NB_Scott Does Object > Path > Offset Path work?

Edit: just saw that Monika posted similar solution.

Offset path can do it.

Object or Effect menu . The latter would need to be expanded afterwards. The former creates an additonal object