By the way, Durham Records Online is a useful source of information via its free index. For instance it is easy to find Martha's burial because the free index includes her address.
https://www.durhamrecordsonline.com/NewSearch.php well worth bookmarking 
I can only see ones I have to pay £1.50
Am I doing something wrong ?
It's a pay-per-view site.
But if you search for a burial of a Martha Isabella Brown in 1944 it appears on the results page (which you don't have to pay for) complete with the same address as in the newspaper article.
The newspaper article stated burial at Ryton*. It has to be at the cemetery as opposed to the churchyard because DRO only has Ryton churchyard burials up to 1867.
Also, I've just noticed, it also says 'meet at cem[etery]
