Icl this is suugo science at it's finest, you also gave another etymology of the word adal
Cad-dal=the white land, but awdal should be accepted as the universal etymology bro as we got al dimishiqi mentioning it to
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اوتل=اودل
"And the Sea of Mandam and the Sea of Aden, and its outlet is between Awdal and Aden, between two mountains, and it passes by the coast of this man called the Gulf of Qalzam to the north, on the A'jam mainland, because the eastern land of it is the Arab Land, and the coast of the A'jam mainland passes through the country."
So adal should be how non-somalis pronounced awdal. Somalis probably didn't pronounce it as adal, we got sayyid Muhammad abdullah hassan mentioning it too (one of our greatest poets) as awdal:
"Bullaxaar agteediyo hadaan, Awdal yaacin"
And one last thing, the locals didn't call the kingdom adal didn't you wonder why the name adal was never mentioned in the futuh?
They called it ba'ar sa'ad al din and there's another account of them calling it zeila
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So yeah awdal most likely didn't come from adal but the other way around nor did carro dir give birth to these terms (doesn't make sense at all).