Generally speaking people in the arab world considers the right hand to be the dominant and proper hand despite the writing system. This is reflected in muslim scripture. For example the hadith states that ablution should begin with the right hand and th right side of the body and then continue to the left, affirming the right side as more pure. "What your right hand possesses" is used as a euphemism for slaves in some places in the hadith, suggesting that the right hand is dominant, commanding.
As for how left handed people were treated İ can't comment directly. Do you have a source though that right hand favoritism is at all related to writing direction?
I'm not sure about your explanation about lefthandedness going against proper writing. What's the source for that? It seems strange to me since many of those superstitions would've developed among people who were illiterate or semi-literate. I don't have a counter-example, just curious about a source.
Anyway, Jewish texts do acknowledge the existence of left-handed people, while preferring the right side of things generally. The right side is seen as better, indicating preference. The bible talks about the sacrifice of the scapegoat, where two goats are picked, and one is randomly picked (through lots) to be sacrificed, the other to be sent into the desert. The Talmud has the goat on the right ideally picked as the sacrifice, since ideally the goat on the right, the preferred one, would be selected. Generally in the bible and later text the right side is the one seen as having an advantage.
Yet, those who use their left hand are not judged for that. In the bible, Judges includes a narrative of a left-handed man using his handedness to launch a stealthy assassination. While the story only makes sense knowing that people generally assume others are right-handed, nothing negative is said about his handedness, and he's a positive character who is left-handed. Later on in Rabbinic texts left-handed people are acknowledged as existing and not spoken about negatively, while still referencing right-handedness as the default position.
In the Old Testament, I think that there are only three mentions of left-handed people. None of them seem to be regarded negatively.
It may be significant that although there were laws on personal hygiene (burying faeces outside the camp, for instance), they did not mention using a particular hand for cleaning oneself or for eating. Also it is unlikely that there was wide-spread literacy in the periods described.