360 magnum troubles

I'll agree that from the symptoms when the throttle is opening gradually, it sounds like it is just going too lean overall. The suggestions on the transition circuits is where I would focus.

Also, having a stock PCV valve with this cam is going to cause problems; the vacuum signal from this cam is going to be low, and a stock PCV will be at the wide open setting all the time, and will dump a bunch of extra air into the intake.

And, I suspect the carb does not have a choke plate on the top of the primaries which is making things worse.

OK on the spark plug gap and ignition. Just to be sure it is good, take one spark plug wire, and place the metal end 1/4"-3/8" from metal and crank a few cycles. You should get a bright, blue spark. If so, then move on to the carb and intake.

Since this is a used carb, look at the butterfly plates and see if there are any small open holes drilled in them. And look at the airbleeds in the top to see if anyone has drilled them out. A used carb is a VERY big unknown here.....regardless of having been gone through. Do you know any history of the carb? Do you have another that is known good?

Some troubleshooting suggestions:
1. Block the PCV and see if that helps. If so, then it is time for another PCV.
2. Spray some combustible liquid around the intake flanges and carb mountand see if it speeds up to see if you have a vacuum leak. (This will not say if you have a vacuum leak on the underside of the intake-to-head flanges.)
3. Get a vacuum gauge and take intake vacuum readings at idle.
4. Cover the carb primaries most of the way with a gloved hand and see if that helps it to run as you open the throttle a bit and gradually; if so, then it is the transition adjustments.

If you are not aware of what the transition slots are, there are some small slots in the carb throats just above the throttle plates when closed. These feed full into the carb when the carb operation is 'transitioning' from idle to the main circuits. The secondary has a a stop that sets how far the secondary butterflies are open when at idle; this setting determines how much air is fed into the intake from the secondary side, and that stop is adjustable. Then there is the adjustment of the primary transition.....

BTW, have you been able to get the revs up to break in the cam? If not, then you ought to stop any extended cranking or idling.