@fsjdw2
OK, Good photos and detailed response. Before I forget, make sure to look through the tube feeding air to the airbox, and make sure a shop towel, mud clod, hornets nest, towel or something didn't get stuck in there.
Also, make sure the choke plunger is free to move in its' bore. This is probably culprit numero uno.In the pic below, this is the big black knob (snicker), but on your quad it's connected to a cable connected to the lever at the handlebar. A kink in the cable or an improper function of the plunger means you're cruising with the choke on 24-7.
You can see the wee return spring. Doesn't take too much to hang this up. You can use a wrench to unscrew the hex from from the carb body, and this will pull out (snicker) with it and then diddle the lever to see if it reacts accordingly. When the little plunger is bottomed out (more snickers), then the choke circuit is closed off, and this ceases the extra fuel flow for startup. If this wasn't screwed in properly on the new carb (boogered threads, bound cable, or other issue), it can make the quad run rich.
If that isn't it:
Process: Identify if black is ignition or fuel related.
MAKE SURE THERE ARE NO FUEL LEAKS. SHUT OFF THE FUEL VALVE BEFORE THIS PART. I have literally burned a bike to the ground because I didn't do this first. With a fresh new plug (I did enough bikes I just kept a new plug in my box), pull the plug cap, stab in a new plug, and while holding the plug body against the engine (so it can ground), crank the motor (the original plug should still in the motor). Your spark should be bright blue, easily visible in daylight (maybe not noonday sun), and you might even be able to hear it. Notice that the actual plug testers have a HUGE gap, but sometimes it's a bit huge for jap bike ignition systems:
Theory: just like the tank on your toilet (which is much easier to watch for illustration), the carb has a float that rises with liquid level, and float controls a valve (the needle in the needle and seat system) that shuts off liquid at the right height.
In your carb, if the liquid is too high, it will run too rich.
Running rich as hell, the other (and far more common) option that produces a black spark plugs condition, can be a few things.
Valve lash is a little more involved. If you have feeler gauges, you can be taught to do it, but it's really more of a shop-level maintenance item. If the valve lash is non-existent, the valve gets held open slightly and you're blowing combustion pressure back down the intake. Carburetors work in both directions, so the fuel gets mixed with air once on intake, again on blow-back, and AGAIN on re-intake, so the bike runs pig-rich. Typically, it's tough to do this because it's a fine line where the bike will still run with poor valve seal. This quad still has full exhaust on it, right?
The other option is remove the carb, and do some looking. You do NOT need to remove the throttle cable.
There are typically four screws holding the bowl on the bottom of the carb. Be careful when you pull the bowl OR turn the carb over, it's going to spill fuel. There's a pivot pin for the float and the presence of the bowl is usually what retains it, so hold the carb level while you pull the bowl straight down after you remove the screws. It'll probably stick, and be careful not to tear the gasket. The pictures below have a screw holding the pivot pin.
Once that's off, identify that pin and keep an eye on it, you won't be finding a replacement for it at Lowe's. Invert the carb, and look at the frame for the float, where the float pivots on that pin. The frame is usually flat sheet metal, and should be parallel to the bowl gasket surface when the carb is upside down (and not tilted). It's easy for these frames to get bent, and if the floats are bent up, they'll close off the fuel supply late, and the fuel level in the carb will be too high, and cause a rich running condition. Photo shows different carb, similar system. Frame is the shiny silver part.
Some places talk about blowing in the fuel inlet and all that crap, skip all that. you're just wanting the needle (under the tang) to seat when the floats are in the right spot, and that's when the float frame/arm is parallel to the gasket surface. if you hold it upside down and slowly lift the floats, you'll see that the needle will seat and stop moving, and if you let go so gravity is on the floats, that's close enough to measure. Pic below shows the float a little high, but remember, it's upside down. There's usually a spring-loaded pin in the needle. If so, you want to hold the float so it's not compressing that pin, and see if it's level when it JUST starts touching that little pin in the float. Some needles omit this and just hook directly to the float.
While you're there, look to make sure all the jets are still in place like above, and write the numbers down on them. If it's a used carb, it's possible some t-rard used a wire or drill bit to 'clean' the jets out, and ruined them. The numbers are listed in the service manual to make sure that Kawasaki-san didn't grab the wrong jet out of the bin during assembly, but it's ALWAYS possible some idiot drilled the jet. The jets should be snug, but if you try to make them tight, they'll just break off as they're brass. Also not found at Lowe's. If they're snug, they won't rattle out,
SOME carbs have washers under the main jet, some don't, and some carbs have intermediate jets and some don't and so on. Your exploded view knows for sure.
Pull the pin that retains the float, and the needle will CAREFULLY come out with the float. If the needle resists coming out, there's your huckleberry, it should be free to move. Look for ANY trash, specs of dirt, rust, ANY PARTICLES OF ANYTHING in the needle area, on the needle etc. THIS NEEDS TO BE CLEAN LIKE YOUR WIFE ON YOUR WEDDING NIGHT. These solids will keep the needle from plugging the hole, which is what shuts off the fuel flow into the carb. A little spritz with carb cleaner here will not hurt anything, but make sure it blows it all out, instead of just moving it around to come back and haunt later.
Let us know what you find here. Pics are great. Pics inside the venturi (the horizontal tube through the carb that the engine breathes through) won't hurt either.