I think you are right. But get is not a natural verb for this kind of activity, 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?

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?