Do seraph crystals drop on the Xbox 360 in the Captain Scarlet DLC on TVHM, or do I need another DLC?
Seraph crystals only drop from the Seraph Guardians in UVHM. Per the wiki:
by killing Seraph Guardians in Ultimate Vault Hunter Mode
In the Scarlett DLC, those guardians are Hyperius and Master Gee.
So to answer your question, yes, you'll also need an Ultimate Vault Hunter Upgrade Pack to play in UVHM. You might already have one or both of these. If that's the case, just play in UVHM, kill Hyperius or Gee, and you'll have yourself some Seraph crystals.