What are the communication procedures for flying from one Class D airspace directly into another Class D airspace?
I have not flown into this airspace. But, based on the location in question, you probably should be on flight following long before entering the Mode C Veil for the overlying Class B. If not, contact the Approach Control listed for the Class D into which you are flying. They will then coordinate your clearances through the respective airspaces until you are handed off to the final destination Class D controller for landing clearance.

Arrivals into KADS have a similar scenario. When getting your ATIS before arrival, you will be informed on the recording to contact approach on a specific frequency, give your destination as Addison, and that you have information A-Z. You are required to do this in order to land in KADS. You can not enter the Class B or the Class D without doing this first. Approach will handle your transition through any airspaces from there.