Question - is it not allowed to give dīkṣā while one’s Guru is still alive?
Advaitadas: if individual Gurus do not find this the proper etiquette then that should be respected, but there are examples of disciples of living Gurus giving dīkṣā - Yadunandana-ācārya initiated Raghunātha dāsa Gosvāmī when Raghunātha was still at home, and later Raghunātha dāsa Gosvāmī even met with Advaita-ācārya, who is Yadunandana-ācārya's Guru in Puri. Advaita-ācārya > Yadunandan-ācārya > Raghunātha dāsa Gosvāmī. Antya-līlā, chapter 6, 242, "When Raghunātha dāsa met them all, Advaita-ācārya was very kind to him". Antya 6 verses 158 - 161, make clear that Yadunandana-ācārya was already the Guru of Raghunātha dāsa Gosvāmī -
vāsudeva-dattera teṅha haya ‘anugṛhīta’
raghunāthera ‘guru’ teṅho haya ‘purohita’ Antya 6.161
“Yadunandana Ācārya was the priest and Guru of Raghunātha dāsa. He had accepted the mercy of Vāsudeva Datta.”
Then, much later -
raghunātha-dāsa yabe sabāre mililā
advaita-ācārya tāṅre bahu kṛpā kailā Antya 6.245
“When Raghunātha dāsa met all the devotees (in Puri, later), Advaita Ācārya showed him great mercy.”
Another example is Śyāmānanda Prabhu, who was deputed to Orissa for preaching. In a village called Rohiṇī on the banks of the river Subarnarekhā, the village chieftain Acyuta and his son Rasikānanda became the disciples of Śyāmānanda Prabhu. One day the news came that Hṛdaya-Caitanya, Śyāmānanda Prabhu’s Guru, had passed away. Shortly after that, Śyāmānanda Prabhu installed Śrī Rasikānanda as the Mahānta of Śrīpāṭa Gopīballabhpur. This is confirmed in Śrī Rasika Mangala. Rasikānanda had received dīkṣā from Śyāmānanda Prabhu long ago. Hṛdaya Caitanya (or Hṛdayānanda, according to Rasika-mangala) passed away later. All this is very clear from Rasik-mangala.
Most importantly, there is no prohibition on giving dīkṣā while one’s own dīkṣā-guru is still alive, in any śāstra I know of.