I do play with a strap even when sitting down. I do keep the guitar up high where it is comfortable for my hands.

When I visit guitar stores and play without the benefit of a strap I tend to often feel the same awkwardness the you seem to be describing, where the guitar tends to want to go off to the side. I have to struggle to get a comfortable position. Off course the convexness of certain parts of me may play a factor in my discomforts! ;-)