I am also descended from Thomas Plummer Addison. He was my great great grandfather. My great grandfather was Arthur Richman Addison. Like you I have had trouble getting back another generation. The SA certificates offer little help. I have his death certicicate which gives no information about his parents.
I do have a little more information though. He was married to Eliza Mary Claxton TOURNER. The Biographical Index of South Australians omits the Tourner. They were married by banns at St Brides, Fleet St London on 20th feb 1830. She was baptised on 27 Mar 1808 at Southwark. She apparently was the daughter of Isaac Tourner and Elizabeth Brickwood (from the IGI). They had 3 other children born before they came toSA. Sarah Elizabeth 1831-1832, Elizabeth Mary 1832-1832 and Robert Claxton 1834-. This info is from the baptism registerSt Andrew Holborn. His address in London then was32 Leather Lane, and his occupation steel plate maker. A london directory mentions TP & R Addison Steam grinding Mills - file cutters grinding & polishers 36 Leather Lane Holborn. When he applied to come to SA in 1838his address was 4 Dorrington St Clerkenwell & his occupation engineer.
I tried to email you, but it was returned. Could you please email me, and maybe we can talk. Diana Elias.