What's the source of these written coordinates against which you compare your own coordinates? And when were these measured?
I am not entirely sure, but I don't think a projection issue is the most likely, as any of the other common coordinate systems in your area would put you off by far more than this (up to a 1000 km due to different False Northings).
You seem to be using Palestine 1923 / Palestine Grid, other potential logical options in your area are Palestine 1923 / Palestine Belt, Palestine 1923 / Israel CS and Israel TM, but all of these aren't close in terms of FE and FN.