Do the voltage tests first. I wouldn't replace the 83 with a solid state unit unless your ready to have all the voltages off. Your 83 can still be good and voltages can be off for many other reasons. I think in the 600A the 5Y3 is for the screen voltage and if it goes bad the tube testing won't work at all or will show a very high reading. I'd have to check the schematic.
Set it up as shown to test the 6L6 tube. Ignore the line set for now and check the plate voltage. I believe it is pins 8 (negative) and 3 (positive) for the 6L6. If the plate voltage is 150VDC +/-5V, than the resistor in the circuit for the meter has changed value. Make sure you use a shunt resistor across the test points if you use a DVM or VTVM when checking the plate voltage or you will get a high reading of around 190-200 VDC as its pulsating DC and not pure. A 100K or close to that will work. I remember my resistor in the meter circuit had changed when I went through it for the first time and I had to change it to get the meters pointer to the right spot. It read low as well. I believe the resistor is located on the pushbutton assembly center/top. You may have to change that resistor, while keeping the plate voltage at 150VDC until the pointer hits the test spot. If you move the positive lead on your meter to pin 4 you should read the screen voltage of around 130VDC. The 5Y3 could correct a problem with this voltage. BTW you need to push the P4 button while taking any of the voltage tests.
Try this first and see what you find.