No. Let Gamma be the graph with two vertices and no edges - the non-abelian free group of rank two - and let g be the commutator of the two generators s1 and s2. Then g is certainly non-trivial, but g dies whenever you kill s1 or s2.
UPDATE:
For an example with a connected graph, let's take Gamma to be the straight-line graph with four vertices a,b,c,d (so [a,b]=[b,c]=[c,d]=1). Now consider g=[[c,a],[b,d]]. Clearly this dies when you kill any generator. On the other hand,
g=cac−1a−1bdb−1d−1aca−1c−1dbd−1b−1
and a well-known solution to the word problem in right-angled Artin groups tells you that g is non-trivial.
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