Does the Adhan remain at its appointed time upon the entry of the Dhuhr time, with only the prayer being delayed for cooling, or is the Adhan also delayed until just before the prayer time if the Dhuhr prayer is delayed for cooling?

It is best for the Adhan (call to prayer) to be at the beginning of the prayer time, so that people may know of the entry of the time. Delaying it until the end of the time may cause confusion and lead some to pray after the time has expired. There is nothing that necessitates delaying the Adhan. The Hadith of cooling (ibrad) does not contradict this, for Al-Hafiz Ibn Hajar explained that Bilal's iqamah (call to commence prayer) never deviated from the Adhan, because the Prophet, peace and blessings be upon him, was diligent in praying at the beginning of its time.