What facilities do you have for powering this thing? Do you have a controlled voltage source - either AC or DC - that can be used to limit the fault current?
As your schematic says "HOT CIRCUIT, BE CAREFUL". If you do not have an isolated AC/DC supply then it might be time to make a "dim bulb" current limiter, and proceed with extreme caution!
... ~4A home made fuse
I'll not ask!
But make a mental note that the fault is mostly likely to be in the primary side. "Popping" fuses suggests a reasonably short circuit on the main high-voltage side - either a direct short-circuit, faulty component short (C605, C611, etc) or the switching transistor (Q601) is turned-on, permanently. Anything other than a direct short would probably be sending smoke signals.
What I've replaced: Primary side: D608 (replaced with 1N4747),
Why? What led you to this?
Also, a quick google suggests that D608, a 1ZB36 is a 36V zener, whereas a 1N4747 keeps returning "20V Zener".
Q601 (replaced with IRF842)
Again, why? Have you determined for certain that the original Q601 was defective?
Ordinarily, targeting the switching transistor and the control IC would be my first line of attack - together with a-n-other faulty component. However, this circuit doe not have a control IC - it's self-oscillatory - so it becomes a bit more critical to replace like with like. I've not checked IRF824
vs TK10A50D - are they identical? Are they close ? I would certainly replace D608 with something closer to 36V operating voltage.
I'm starting to feel like the transformer has gone bad, because when I probe 6 and 2 in continuity mode, it indicates a short circuit
In or out of circuit? If it is still in-circuit there could be something else causing the fault. If it's out-of-circuit - there's your problem.
I'm sort of tempted to just completely remove the transformer, and see if the fuse will continue to blow. Will that yield any important results if I do that?
I suppose it will remove suspicion from some of the other components (like C605/11 etc) - but you could realistically (and more importantly, easily) test these components individually.
My focus of attention would start with Q601 - is it short circuit? - Does shorting gate and source allow power to be applied without the fuse blowing? No - you've got a short that should be traceable with an ohmmeter. Yes - move on to Q602. It is open? - remove and perform diode-test between base-emitter, base collector, etc. Not a conclusive test but if Q602
appears to be good, look for things that are holding its base down - like Q621.