craftsman snowblower will only start with choke off instead of on--is this a problem

I am new to the forum and glad to be part of this community. I have a Craftsman snowblower Model 536886540. Here is my predicament. My engine would only run with choke placed in the Full position. I went ahead and removed the Main Adjusting Screw Assembly and cleaned the carburetor bowl, the jet to include both of the holes in the jet until carburetor cleaner came out of both holes, and then installed the bowl and Main Adjusting Screw Assembly.
 
So now, the engine will not start with the Choke in Full position but will now only start in the Choke Off position with throttle control in Run position. I adjusted the carburetor according to my user manual.
 
Please, if you have any thoughts to help me out, I would sure appreciate it. Perhaps this is not a problem at all. I do not know. The manual says that to start a cold engine to follow the instructions below.
 
1. Move Throttle control to Run position
2. Rotate choke knob to Full choke position
3. Push primer button two time if temperature is 50F to 15F which is always the case.
4. When engine starts, release starter button and slowly rotate choke to Off position
 
Thank you all in advance.
 
Zenny