I'm coming late to this but I want to answer a few points

1) south Wales speakers of Welsh would be aware of the 'standard' form of 'gan' i.e. gennyf etc. These forms are taught in schools, they are in normal literature and they are actually still used in the south in different circumstances to 'gyda' i.e. 'cafodd ei weld gan John' - 'It was seen BY John' here 'gyda' would not be used.

2) I don't think anyone would say 'cyn imi egor fy mab cyntaf' - firstly 'esgor' is quite literary these days - it would literally mean 'before I gave birth to my first son'. The way that one would normally talk about having a child (in the sense of a child entering your family, not 'owning' a child) would be to use 'cael' for example, when my wife had our child one of the first questions was 'Beth gawsoch chi?" (what did you have?) the answer was 'Cawson ni fab' (we had a boy). When I say 'before we had Eoin' (his name is Eoin) I say 'Cyn inni gael Eoin'. This is the standard and normal way of saying this in Welsh.

3) Esgor is followed by 'ar' so, Esgorodd hi ar fab 'she gave birth to a son'. However, one has to understand that you don't really hear 'esgor' very often unless one is actually talking about the physical process of birth.

M