Yes, the cam sprockets are different. That is what is keeping your engine from running properly. You will need the driver side cam sprocket from your old motor installed on your new motor.
As you implicate, the FSM has dial indicators, special cam locking tools, remove a bunch of junk, etc, etc
Swapping cam sprockets is such a rarity that I haven't seen any tricks like we have for replacing the timing belt.
I would guess that if this timing mark (notch) on the sprocket represents the same "thing" on all driver side cam sprockets, then you might devise a method to shortcut all the hassle in the fsm.
Maybe if you could lock the driver cam down somehow so no movement is possible, then you could simply make a very accurate mark of where the 04 cam sprocket is at(like in the picture) and then swap in your older one. Line up the old sprocket to the mark and then tighten a brand new cam sprocket bolt. I could only envision using a the dealer cam lock tool that fits on the back of the cylinder head. To even try this shortcut, you would have to do all the procedures used for removing/replacing the timing belt.
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They will usually start and run if you unplug the cam sensor...if you forget to change the sprocket that is...but wont run good, and not over 2500 rpm...they will not start at all if the flywheel is not changed.