Asking my PhD advisor for a postdoc position. Is it bad to appear desperate?
First of all, your advisor of all people really should not be somebody who you need to play games with regarding the state of your career (academic or otherwise). They really can only do their job (advising you, also in questions of career development) if you are being honest with them. In that sense, it shouldn't really matter if it "looks bad" - if your situation is bad they should know.

Further, your advisor already offered you a temporary job, and presumably they did so under the assumption that it would only be a fallback if you did not find a more permanent next career step. Now this situation has arisen - why would it be bad to take them up on their offer in exactly the situation that the offer was made for?