Hello Tessy.
Both phrases are correct.
I am dead sleepy. - In this case, ''dead'' has the meaning of ''complete(ly), very''. For example, ''The exam is dead easy. I am dead hungry.''
I am sleepy to death. - In this case, ''to death'' refers to ''extremely''. When somebody is bored, frightened, sleepy, etc. to death it means that somebody is extremely bored, frightened, sleepy, etc.
I hope this is helpful :)