Suppose we ask the following question:
Did you know I'm getting a new job?
Does it suggest that I've already found the job but probably not started yet. I mean can this meaning be possible in some context?
I think you are right. But get is not the best verb for this kind of activity (though it is often used), and as you surmised the true meaning is not so clear.
As you mentioned, found is probably a better verb to use:
Did you know I found a new job?
If you had started the job:
Did you know I started a new job?
And if you had not found a new job yet:
Did you know I'm looking for a new job?
Did you know I'm getting a new job?
Does it suggest that I've already found the job but probably not started yet? ~ No, it does not!
But for the Title of the question: Present progressive for the future event - YES, it's possible.
To mean that you already have a job but not going, you need to clarify that
Do you know I already have a job... (but for 'x' reason, I haven't joined yet...or whatever).
I'm getting a new job talks about the process. Maybe, you are selected and about to get a 'joining letter'.