An engine that needs partial choke to run smoothly means it is running lean (too little fuel, or too much air, or both). Most often, it is restricted fuel flow due to slime/gunk in the main jet / nozzle area, and this can happen with fuel that is more than 30 days old.
Was the new carburetor a genuine Honda carb, or aftermarket?
There are a number of gaskets, insulators, etc. when fitting a Honda carburetor, including one with a tiny vent line. They can easily be installed backwards or upside down. Double check these, as well as the correct holes for the governor rod and throttle return spring.
Be sure the choke is operating properly. Here's an image from the GX390 shop manual to help:
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Caveat: I work for Honda, but the preceding was my opinion alone.