There is restriction of fuel flow somewhere. Look for a clogged fuel filter, a pinched fuel line, or debris in the fuel tank. Otherwise, the carburetor needs to be re-set.
Your Briggs and Stratton 5.5hp only runs with the choke on because it needs a tune up.
There is an arm that must go to the choke and I'm not sure where. I think the plastice piece might be broken.
There are several systems used by Briggs for choke control. Auto choke engines are equipped with a heat sensitive bi-metal coil attached to the exhaust. When cold, the coil closes the choke plate. When heated by exhaust gases, it opens the plate. Most other models have a manual lever to open and close the choke or a primer bulb system.
check that your main jet is not clogged. The reason it seems to run better while choked is simple. there is a bypass tube that siphons extra fuel when the choke plate is closed
Clean the carburetor with special attention to the high speed jet and fuel passage.
The choke has to be half closed to run for two main reasons: 1. The carburetor linkage is out of adjustment. 2. Fuel flow is somehow restricted from flowing freely. This can range all the way from debris clogging the outlet of the gas tank to clogged passages in the carburetor. Could be a pinched hose or a clogges fuel shutoff valve, too.
When you say the engine will turn over only when in choke, I assume you mean it runs while chokedand then dies when you let off the choke. Check the fuel filter,the air filter,makecarburetor adjustments if necessary and ultimately, checkthe throttle linkage.
It could be that the spark need to be replaced. Other reasons could be that the carburetor needle valve is opened too far, or the float is stuck in the open position.
Check to made sure the choke is fully closing. It's possible the choke cable could've slipped preventing it from choking enough. By the same token it could be closing just enough to not cause a problem during warmer weather.
Small objects, like small fish bones, can cause you to choke.
The kill-switdh wire is disconnected. loose. or broken. The kill-switch wire starts from the OFF position of the ignition switch and meets another wire that leads to GOUND to stop the engine.
Try cleaning or replacing the air cleaner. A clogged air clearner will make the engine run rich (like having the choke on). The rich mixture will cause the blackish smoke. The fileter may not even look very dirty, but the element can still be plugged and cause a problem similar to what you describe.