Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear brother in Islam, we would like to thank you for the great confidence you place in us, and we implore Allah Almighty to help us serve His cause and render our work sincere for His sake.
In his response to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states,
Islam is a religion of balance and, as such, there is a time for everything. At the time of Salah (ritual Prayer), there is no dhikr (remembrance of Allah) better than Salah. Likewise, at the time of the Adhan there is no better dhikr than repeating the words of the Adhan after the mu’azzin. Once the Adhan is over, we should offer the supplication prescribed for the occasion as has been reported in the sources.
It has been reported by both Al-Bukhari and Muslim on the authority of Abu Sa`id Al-Khudri that the Messenger of Allah (peace and blessings be upon him) said, “When the muezzin calls to Prayer, you should repeat the words after him.” In another Hadith, it is reported that the Prophet (peace and blessings be upon him) advised us to say the following supplication at the conclusion of the Adhan,
“Allahumma rabba hadhihi ad-da`wati at-tammah wa as-salati al-qa’imah ati Muhammadan al-wasilata wa al-fadhilata wa ib`athhu maqaman mahmudan alladhi wa`tahu.”
“O Allah, the Lord of this perfect Call and the Prayer we are about to offer! Grant Muhammad the privilege (of intercession) and honor, and raise him up to the eminent position that You have promised him.”
We further read in another Hadith that we may then engage in any supplication or prayer we may wish afterwards and also that the prayers offered then are worthy of being answered by Allah.
To conclude: We are not supposed to engage in the dhikr you mentioned instead of responding to the muezzin, but we may say them afterwards. May Allah help us all to adhere to the Sunnah of our beloved Prophet (peace and blessings be upon him), for our salvation rests on it.
Almighty Allah knows best.
Excerpted, with slight modifications, from: www.muslims.ca