Should I adress myself as a doctor (Ph.D.) when I am only waiting for the diploma to be signed?
As I understand it, you have met the requirements for graduation but you have not applied for graduation yet, so you have not graduated. The degree of Dr. has not been conferred to you by the university, so you should not use it. 

The importance of this varies by country: in some countries such as Germany, this is explicitly stated in the university regulations. In the US, many universities have three official degree conferral dates every year (at the end of each term, see for example https://web.iit.edu/registrar/academic-calendar), and you are not expected to use the title before the degree has been conferred.

Others may address you as they wish, but trying to use the title yourself might be seen by some as dishonest.