Thanks for the responses. I did some reading up on holding the guitar classical style and decided to just stay the course, use my strap and keep my back straight.

It's funny that haghj500 mentioned using a non-slip mat. When I was first messing around with the different positions, I played around with a mousepad I had lying around to see if I could keep the guitar from slipping down my leg. It didn't really work.

I have short legs and discovered that I need my knee slightly elevated to get the top of my leg to slope back into my body in order to be comfortable playing without a strap. A stool with a good cross bar works.

At the end of the day, I prefer using a strap. It allows me to hold my guitar in a semi-classical position with the guitar body centered on my torso. The only difference is that I don't rest the guitar on my leg at all, it is suspended by the strap. This is really comfortable for me both sitting and standing.