We have not tested the FZ82, so I cannot say if it has a live/clean HDMI output. I can see in the Manual for the FZ82 it has a number of settings for both Recording mode and HDMI output. Make sure both of these settings are in 1080p60.
As for the Sony, do you mean the HX400? If so, the same as the FZ82, we have not tested it but it has the same settings I mentioned above for both Recording mode and HDMI output. Check that they are both in 1080p60 mode.
As a backup, try settings both Recording and HDMI output modes to 720p60. Also, try connecting directly to a TV and confirm it does offer a Live, Clean, HDMI output with Audio in 1080p or 720p. If this is true on a TV, it should work the same way on the Webcaster.