I got around to doing what I should have done first and just looked down at the engine mounts. Looks like the driver side mount has completely separated. It's hard to see the passenger side one but it doesn't look like the top nut is centered on the mount so I assume that it is broken too. I did get a photo of the driver side mount (see below).
The mounts don't seem too hard to replace (any advice appreciated), but is the fact that the engine is skewed to the left a little going to be a problem. I just got the transmission replaced - could there be any issue with how it is installed that might cause the engine to be positioned wrong and break the mounts?