Ok that might be the problem because I got a new throttle cable. I'll check that out. Also, when I took the slide out of the carb to put the new carb in, it was wet with gas. Does this mean anything?
I also still don't understand why it would die when you apply throttle...

 

I fixed that issue. The throttle cable was the wrong size. I'm using the old carb because the threads on the new one are stripped. Now It will run but after a few minutes it will rev up and then die. After that, I can't start it for about 5 minutes. When its idling it will go up and down a lot and it is not a steady idle. It wasn't doing any of this (except it would bog down a lot!) until either I messed with the air/fuel and idle speed screws or it was because the atv got wet from the sprinklers.

 

Also I can only get it running when the air/fuel mixture is as rich as possible. This is because the choke lever is blocked by the frame so I can't use it. I was riding the atv around the yard and it worked fine (there are some big bogging down issues) except when I put it in neutral it did the weird rev die thing. Also, after pulling out the spark plug after riding, it was pitch black. Is that because it's running so rich or something else?
Another question:
I have a pz20 carb on the ATV right now but the choke lever is blocked by the frame. Would it be fine if I installed a PZ19 carb which has a different style choke lever?

 

If the plug is black, it is too rich. Don't forget you have two mixture settings, main jet and idle jet both have to be set right. Check that plenty of fuel is getting to the carb, a rise in revs followed by stalling, is what happens when you run out of fuel. Using a 1mm smaller bore carb won't make much difference, in theory it will produce less power at wide open throttle, but most engine makers "over carb" anyway and the amount of gas getting through the inlet valve, at wide open throttle, is the limiting factor.

 

Ok, I'll probably buy a PZ19 carb. I think that there are some fuel flow issues as the gas tank is barely above the carb if not at the same level. Once I get the new carb I'll be able to start it with the choke and then get a good air fuel mixture. Would a better air fuel mixture make it have less bogging down? If not, what should I do to get less bogging down and how do I get a better fuel flow?

 

Check the fuel tank filter if it has one and fill the tank, that should ensure enough to give good fuel flow. If there are O rings or gaskets between carb and manifold, make sure they are fitted and sound. Don't adjust either main or idle jets unless you have to and the engine needs to be at working temperature if you do adjust them. Idle speed (throttle stop) will almost certainly need adjustment though.