I guess it is better to measure when shadows are longer, but there's more math to do.

At 12 PM June 21 at Houston, the sun's angle of elevation should be 74.6 degrees if round. If flat, and using the numbers of the flat earth model, the angle should be 83.6 degrees.

At that time, the sun was at 111 degrees 7' W and at the tropic of cancer. I figured out the distance between Houston and the point where the sun is directly overhead, and from there I figured out the sun's angle for the round earth.

With the flat earth angle, I also figured out the distance between Houston and the "sun" point except using the figures for the diameter of the flat earth and the height of the sun.