Hi Andy
Just to tackle a small bit of your query

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You refer to the death indexes starting in 1813. Did the ones you looked at come from Family History Online? If so, they start at 1813 because they're far and away the easiest to transcribe, so most family history societies start with those.
From all indications (Genuki etc) it does look like there will be burial records from Awre/Blakeney from long before that, although the earlier you get the more likely that there's missing pages

The Gloucester Record Office is the best bet, but you're not limited to email. Quite a few RootsChatters go there on a reasonably regular basis, so could look up Jane no. 1's death for you. I should be going myself, but it won't be until after my (immanent!) holiday.
I think you're right about the missing marriage. Where the LDS have transcribed parish records they are reasonably complete. I would normally think that Jane Hawkins must have come from elsewhere and therefore married in her home village, but if she had a child in Awre prior to the marriage, it seems unlikely that they would have married elsewhere. It's probably worth looking in Newnham though. I would take the IGI references you have with a distinct pinch of salt - I've had a look and they seem to be guesses to me

The fact that Jane's baptism doesn't seem to be there might also be confirmation that she came from outside (or she was a widow?)
I think it just has to be the parish records. Can you get to an LDS family history centre and order the microfilm to look at yourself? They usually (but not always) have surrounding parishes on the same film so you can kill more than one bird with one stone.
Best I can suggest at the moment!
Best wishes
Angela