So I checked the continuity between the coils and the case, and there is nothing. The coil measures 17.5 ohms resistance. Any thoughts?
Assuming your line voltage is 220VAC, the coils will draw 12.6 amps at 17.5 ohms.
I am only asking to be sure, but have you closely checked the fuses to assure they are 20 amps rated? Have you tried several ones to assure it isn't a fluke. Did the fuse blow violently every time? I would stick in fuses from another box of fuses and see if that makes any difference.
Also, have you tested the unit with the control box cover with the SSRs not closed up. Something may be pressing against another component or a ground when the covers are installed. If that is the case, everything will test fine until you close it all up, and then it won't work.
Some ideas for testing ( test with the cover off the control box):
Disconnect the coils at the SSRs and turn the power on only ( the PID off) - did the fuse blow. If it blew on this first test with nothing on but the main power switch, you have the switch wired wrong ( or a bad switch) ... which is what I am starting to suspect.
If it did not blow, turn on the PID, but don't run the heat program - did the fuse blow?
If not, run the program- did the fuse blow?
If not, then the coils are the culprit. At this point you have run everything in the circuit with only the coils removed.
To verify that the coils are truly the problem, connect a 100 watt light across one SSR and turn the system on - did it light and the fuse not blow? If so, change it to the other SSR. Did it still light and things are OK. If both those tests are good, there has to be a short or other fault in the coils.
If the fuse blew at any stage of these tests, the problem lies between that test and the last one.