Generally what you describe is due to a badly-tuned carb or timing issues, but I'm confused about the cam now. Certainly if your method of finding TDC was off a bit, you'd probably wind up with loose rockers instead of overtight ones. But I don't understand why you wanted to reset them with the motor running. Are you using solids? In that case you would never do the 1/4-turn thing at the end, you'd set them with zero lash and then adjust them while running (hot) with a feeler gauge. Assuming you have hydraulics, it is bothersome that you had some that wouldn't show lash until a full turn, but that still doesn't mean you trashed the cam necessarily.
All of this is why I went to roller rockers long ago. You can fingertighten the polylocks, then give 'em the additional turn (I go 1/2 turn). But you don't zero-lash them by going down until the pushrod no longer rotates, just until the rocker is seated. You can force the plunger down too far otherwise.
The fact that you just did 1/4 turn at the end helps somewhat. I would check my timing again, and try another carb to be sure that's the cause. As for the cam, try the method I suggested (again assuming hydraulics). Loosen them all up, go down until the lash is gone (the pushrod can still be turned usually), then go another 1/2 turn. You should have no problem with overtightening, though you can check for undertightened, which could happen if your cam wasn't totally on the base circle when you made the adjustment.
Another advantage of aluminum rollers... they restrict oil flow so you can watch 'em running without taking a shower in hot oil.
So, just to be orderly...
1. Reset the valve lash as suggested to be sure you aren't grinding yourself a new cam
2. If it still burps, look for easy stuff like crossed-up plug wires, timing etc.
3. If it still burps, swap carbs and see if that works.