If you look down the intake side of the carb, there should be a small metal "window" and that will flutter back and forth, that is normal. Then you should have your choke lever and that should move a larger metal "door" that also has that window as part of it. What you want to check for besides your carb is wether or not your airfilter is dirty. If the filter is dirty, then you will have more fuel than air than you normally should have. Also a carb rebuild kit will have new fresh gaskets for the carb. Even though all your gaskets may look ok, they are possibly as old as the trike and simply all squished down and dried up and can let air in sports you don't want air. When you had your carb apart you should have checked you idle jet and main jet to make sure they aren't clogged. Not only do you want the middle of the jets not clogged up which is easy to see, but you don't want the little holes on the sides of the jets clogged up either and they are way harder to tell that they are not clogged up. Also you want to check where the "e" clip is on your needle that goes down into your main jet. It should be in the middle of the 5 rings. If it is in one of the lower 2 rings, it is probably letting in too much fuel too soon. You also need to make sure you have good gaskets in between the carb and the intake to the head and between the intake to the head and the head. In between the intake to the head and the head, there should be an plastic insulator as well as an intake gasket. The insulator is there so the intake doesn't get to hot from being in contact with the head. On the back of the carb, there should be a place for a rubbe o-ring, that should be in place. I make sure that I run the o-ring and 3 gaskets between all the mounting points where there could be an air leak. It's a little overkill, but it beats banging your head over a small air leak which may make your trike run all kinds of crazy. Also on your idle needle and your fuel and air needle, there should be small o-rings, if not, that could be a problem. A carb kit will have all the o-rings that you should need, the intake gaskets will have to come from a engine gasket kit or you will have to buy some gasket material at the local auto parts store and you can make them your self with a marker and a sharp knife if you take your time. You also could have valve adjustment or timing issues but one step at a time.
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