Statement I: Potassium dichromate (K$_2$Cr$_2$O$_7$) is indeed commonly used as a primary standard in volumetric (titrimetric) analysis. It is a powerful oxidizing agent and has a high equivalent weight. Furthermore, it is readily available, it can be obtained in a high state of purity, it is not hygroscopic, and its solutions are stable. However, the reason it is preferred over sodium dichromate (Na$_2$Cr$_2$O$_7$) is not due to its solubility, but rather its stability and molar mass.

Statement II: Typically, the solubility of salts increases with the size of the cation. Sodium ions (Na$^+$) are smaller than potassium ions (K$^+$), so sodium salts tend to be more soluble in water than corresponding potassium salts. Thus, we would expect sodium dichromate (Na$_2$Cr$_2$O$_7$) to be more soluble in water than potassium dichromate (K$_2$Cr$_2$O$_7$).

Therefore, the correct answer is Option A: Statement I is true but Statement II is false.