I think this is a tuning or setup problem that may or may not be fixed with parts replacement.
A 600 Edelbrock will be happy as a clam on a 390, even a healthy-ish street one. It won't make the same power as a equal Holley, but by no means should it be choking the engine and going pig rich on the plugs and blowing black smoke. In fact, they tend to do very little wrong other than being down a little bit on power
First - Make sure the air cleaner lid isn't slammed against the top, filter isn't plugged, make sure no linkage is binding the choke, and the choke is working properly. It needs to be open when hot and not too tight when cold, plenty of videos on You tube. Even if adjusted correctly, it needs to be working too, wide open warm, and not able to close under acceleration. Choke thermostats do go bad, and wiring can be incorrect letting it close on you
Second - I'd put the carb back to stock with a full rebuild, carefully make sure all adjustments are right and linkage doesn't bind on intake etc. Don't assume anything, and be gentle with the parts, check all the numbers of parts, make sure it hasn't been modified
My hunch tells me those two things will do it, however, next make sure the engine has heat, run a thermostat. Then look at the ignition, what it is means less than if it's working right. You should have a mean blue spark from an HEI, if it is anyhting less than that, look closely at coil, module, grounds etc. The ignition could be bad as a small change in a/f generally does not bring you from pig rich to surge FWIW, HEIs can generally handle it, but I tend to not go big on gaps, although some claim wild benefits, typically, .035 is easier on ignition parts and will fight you less with misfire on a marginal ignition. Likely NOT your problem, but my technique is .035 for most builds
I could see you buying a 40 dollar carb kit, or a coil/module, but I really don't see this being a brand/parts mismatch, requiring a new carb and ignition, you are simply having a tuning or malfunction problem with the parts you have