Battlewyrm Carapace drops from Archaeoaevis and Skulwyrm in Zertinan Caverns. The latter can also be found in the upper layer of the Great Crystal. They can also be stolen and poached from the Archaeoaevis. You can also poach them from Dragon Aevis in the Stilshrine of Miriam and Emperor Aevis in Paramina Rift.

Adamantite drops from Adamantitan in Cerobi Steppe, are stolen from Thalassinon on the Phon Coast (rare monster), and are given as a reward for the Darksteel mark.

Just, uh, there.