Number 3. Think of 'chores' as a synonym of 'housework'. If you mean a specific chore, it's more natural to name it. I'm busy doing chores/housework (in general) or I'm busy doing laundry (specifically). We can also say anything is a chore to mean we find it tedious and annoying. e.g. I used to love going to parties, but now I find it such a chore

I'm busy doing chores. I'm busy doing some chores. I'm busy finishing up my chores for the day. I'm busy with some household chores right now. (cleaning, tidying, etc.)

I agree with Claire that the third is the most normal sentence: I am busy doing chores. This means that you are doing jobs that need to be done in and around the house - cleaning, tidying, laundry, gardening etc. By the way, this is a US English expression. We don't tend to use the word 'chore(s)' with this meaning in British English.

Thank you

I am BUSY, not busing.