My weakness

I tend to have too much concern when I work as a member of a team because I want my work to be done properly. To overcome my weakness, I've made a habit of communicating with co-workers consistently (constantly?). This enables me to be more careful in various situations.

----

Nice writing. There is nothing in here that is grammatically incorrect. That's very good!

I am not sure whether you wanted to say "constantly" or "consistently". The two words have slightly different meanings, and I think constantly is better in this situation. To illustrate the difference: 

"I deliver consistent results" --> The quality of my work is the same every time (good OR bad)

"I deliver constant results" ---> I am ALWAYS delivering results, every day (usually good!)