No, there need not be such a geometric triangulation.  Here is a construction.

Consider $A$, a flat annulus, of width $W$ and length $L$.  Here we assume that $W$ is very large and $L$ is very small.  (That is, take a $W$ by $L$ rectangle and glue the long sides.)

Let $\alpha$ and $\beta$ be the components of $\partial A$.  We glue many sub-intervals in $\alpha$ to isometric sub-intervals in $\beta$, via some complicated permutation.  This gives a high genus surface $S$.  The singular points all live in a graph -- namely the image of $\partial A$ after taking the quotient. 

Note that the diameter of $S$ is at most about $(W+L)/2$.  However, any geometric triangulation must have at least two edges crossing $A$ and these edges have length at least $W$.