At the top of each of your posts is an empty div tag. Inside that tag is a style attribute with a clear: both rule.Turns out that in Blogger under Settings --> Formatting there is a setting for "Enable float alignment".
That is what is causing the problem.
That setting "Allows image and text alignment options using the <div clear:both> tag. (Choose "No" if you are having post layout problems.)"
It was turned on, now it's turned off and things work right again.