First, let me tell you I'm totally clueless, and that also covers using GC to measure vapour pressure

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If you are using a standard method, you should check that all of your equipment is operating correctly and leakfree, and perhaps contact your instrument supplier or method authors for comment about your problems.
If I understand your methodolgy, I'd be quite concerned about your use of a syringe, at the least I would be using a 6 port gas sampling valve, possibly in heated valve compartment set to the desired conditions to ensure the equilibrium was maintained during injection.
I can't see how the syringe would be accurate, especially if at a low pressure, as the moment you inject it, it would be at a different pressure, and also would not be swept empty. You possibly could use a headspace sampler, if available.
Bruce Hamilton