So a quick update:
This weekend I pulled all of the socketed chips in the video section and inspected the chips and the sockets.
I replaced the 48-pin DIP sockets for Paula and Denise, as they were both corroded. The PLCC socket for Fat Agnus was also replaced as a precaution.
I am still seeing some weird sync issues. Does anyone know what this part is:

Vsync is 60Hz as it should be, but Hsync in 31KHz mode is measuring at 48KHz!
It looks like a potentiometer of some kind, and it is heavily corroded on my board. The sync jumps around if I try to adjust it.
Looking for the sync adjustment in the wrong place, the thing you are looking at I believe if I remember right is a fuse for the clock I think. The sync you want to adjust is in the back of the Amiga between the switch and the VGI connector, HOkay
smerf