May not be a stuck float. You may have an intake leak at the rubber carb mounts. Typically that will cause that symptom. The rubber mounts harden over time and if the clamps are not seated and tied properly, the motor will suck air there.
Removing the carbs with the air box in place without tilting the motor is next to impossible. Do yourself a favor and lower the motor. You save time in the long run and it is not that difficult. If you loosen the exhaust pipes in the back, you can carefully tilt the motor without removing the exhaust. The CBX is not a particularly easy bike to work on, especially for a novice on the bike.
The manual will tell you to tilt the motor. First, it says remove the exhaust, loosen the rear wheel and give the chain as much slack as you can. Disconnect the oil pressure sender, remove the tachometer cable, place a jack under the engine and loosen all mounting bolts, except the lowest one. Watch for the odd-shaped spacer above the chain when you remove the second bolt from the bottom. Lower the engine and remove the airbox. Take the carbs off. When re-installing, new boots will help or heat the old ones up with a hair dryer to make them more pliable. Use a set of large wood clamps to help pushing the carbs in. Do not use a hammer or such to pound them in place.
