My 2005 Honda Shadow 750 starts on one cylinder. If you pull out the choke, it sounds like it is running on both cylinders. If you put it in gear and drive off its on one cylinder until you rev up on the gas, then both cylinders come on, and after it is completely warmed up it runs on both cylinders fine. What is the problem?

Yes, there IS a clutch safety switch.

http://www.bikebandit.com/houseofmotorcycles/2005-honda-shadow-aero-750-vt750c-handle-switch/o/m151126sch409843

Item #8

Pull the plug out and test it on the frame. If no spark you need new plugs. If it does spark look at the plug. If the plug is black and wet it is getting to much fuel and flooding. You will need to adjust your carb's air to gas mixture.

check for power from the coil, sounds like its bad

Some thing you might try is that you have a bad battery or a loose connection behind the head light housing

Your cycle runs when you pull off one spark plug wire per cylinder. There are two spark plugs per cylinder and you did not remove both . Therefore you still have one spark plug on each cylinder firing to run the machine. Your motorcycle is also equipped with a side stand safety switch. Your bike will run on the side stand . But, if you put the bike in gear with the side stand down it is designed to turn off the motor. This is so you do not drive away with the stand down and loose control of the motorcycle on your first left hand turn. Get in the habit of putting your side stand up before putting the bike in gear and your problem is solved. Read your owners manual.

Make sure the kickstand is up before going into gear, it is like that for safety, she will shut down if it left down.

http://www.bikebandit.com/houseofmotorcycles/2005-honda-motorcycle-vt750dca-shadow-spirit-carburetor-component-parts/o/m2904sch406961
Not to sure about this one, ok, if the choke wont stay out, then try to trace the choke cable to the carby, have a close look and see which parts move on the carby when you activate the choke, some thing just needs to be tightened up, so that the choke stays on. Also, you say the bike stalls and runs rough, is this with the choke on? If yes, then that is just a sign that you dont need so much choke on. Just try half choke. Do you live in a warm climate? if so, then you dont need much choke . So there is no adjustment really, only by the amount of choke you apply. Also remember, never use the choke if the engine is already warm. And, only use the choke for the first start of the day, when the engine is stone cold. The correct way to to start a cold bike engine is, pull the choke on fully, crank over engine, but DO NOT give it throttle, (this defeats the purpose of using the choke), when engine fires up and starts, slowly push the choke in till a smooth, high idle is acheived. You should be able to ride off slowly, and push the choke off after about a mile.

my 2001 just had that problem. Take off your air cleaner and look in the carb. There is a plastic venturi there that sticks. it will free up with your finger

Know that is not normal. It should stay out. Yes you may need to get a new one.

Have you tried to adjust the tickover screw on the carbs?