If $E \neq Q_p$ then there may be more $1$-dimensional crystalline representations that the ones you mention. By Lubin-Tate theory, every character of $G_K$ can be written as an unramified character times a character of $O_K^\times$ (after making proper choices and identifications). The algebraic characters of $O_K^\times$ are then crystalline and if $E$ contains $K$, then these provide examples of crystalline characters.
Here is what it basically says, using the same identifications as above: assume that $E$ contains $K^{gal}$ (it's usually harmless to assume that the coefficient field is large enough). If $s$ runs through the set of embeddings $s : K \to E$ and the $a_s$ are integers, then $x \mapsto \prod_s s(x)^{a_s}$ gives rise to a crystalline character of $G_K$ and they're all of this type times an unramified character.
If $E \neq Q_p$ then there may be more $1$-dimensional crystalline representations that the ones you mention. By Lubin-Tate theory, every character of $G_K$ can be written as an unramified character times a character of $O_K^\times$ (after making proper choices and identifications). The algebraic characters of $O_K^\times$ are then crystalline and if $E$ contains $K$, then these provide examples of crystalline characters.