Extent of the Persian Gulf
You are pretty much spot on in terms of the date.  According to the article Shoreline reconstructions for the Persian Gulf since the last
glacial maximum (Lambeck, 1996) based on models of glacio-hydro-isostatic effects of the end of the last ice age and the article Early State Formation
in Southern Mesopotamia:
Sea Levels, Shorelines,
and Climate Change (Kennett and Kennett, 2006) indicate from archaeological observations that at around 6000 years before present (~4000 BCE) that the present shorelines were approximately shaped.  Both articles record an initial inundation of the Tigris-Euphrates delta.

The process is summarised in the image below:



Image source: University of California - Santa Barabara