Grounding questions

+1 on yes, it should be grounded. Will prevent shocks and improve surge protection. May not fix your hum.

At 63 years old, it is time to bite the bullet and come up to code my friend. In panel surge protectors, arc fault breakers and better distribution of your power circuits for the future, not to mention god only knows how many of those fuses are the wrong size.

I have a quote for a new Siemens box. It’s way out of my budget and comfort zone. I’ve been in the home for 24 years with no electrical issues. There is a new supplemental box installed when I had a bathroom reno in 2014. It runs the sump pump in addition to the bathroom circuits. 

Just as a test drop the ground on the amp. Then do the same to the pre-amp. 

Hum independent of volume can be a sign of old/dry power supply caps. I would, per maxwave advise, take it to a friend and try it there.

Also, since you have balanced connection between pre and amp, check where shield is connected on both ends. It should be connected to shell/chassis and not to pin 1 (that would cause noise currents induced in shield to flow thru analog grounds causing constant hum). This is very common problem (1994 AES paper, Neil Muncy).