Just did the light bulb test. I didn't bother digging out a 9 volt battery from the closet, just wired in a lamp through the SSR with all power off. As expected, the light worked, so I'm concluding the SSR is indeed fried closed. Ordering a new one now.
My built in digital ammeter also doesn't work correctly, reading 0.1A without adding load other than the LED lights and PID, and when adding either the element or pump, the meter simply oscillates in the tenths of an Amp. I'm going to go ahead and order a new one as well....
Thanks for the hints!