Have you found Titanium anywhere in your current world? If you have Titanium it means your world will not have Adamantite spawn naturally within it.
When you break the Demon or Crimson altars starting in Hardmode there are 6 possible ores that can spawn, but you will only get 3/6 in your world.
The possibilities are as follows:
- 1st Hardmode Tier: Cobalt or Palladium
- 2nd Hardmode Tier: Mythril or Orichalcum
- 3rd Hardmode Tier: Adamantite or Titanium
For each tier listed above, you will get one of the ores to spawn in your world.
The only way to get Adamantite if you have Titanium as the main 3rd Hardmode tier ore is to fish. Through fishing you'll get crates that have a chance to have Adamantite. Gold crates have the best odds of giving a significant amount of an ore.