If, as you say, she died 'shortly' after John, have you discounted this possibility:
Margaret Dodds, aged 51, March quarter 1919, Morpeth 10b 802.
OK I know! - the age is not quite right - but the informant might not have known her correct age.
I think this and the one diddy gave you (notwithstanding the fact that you are sceptical about the extra initial) are worth pursuing with Morpeth Register Office. Consider asking them to do a reference check, and only to issue the certificate if the person concerned is the widow of John Dodds.
Jennifer