My ggg Grandmother was Mary Baxter b. approx 1832 Lanarkshire to John Baxter - Ayr 1806- 1883 and Elizabeth Jamieson - Stirling 1808-1862
Can you please let us know where your information comes from.
There seem to be conflicting records.
According to the 1861 census in Hoddam Dumfriesshire (Ancestry transcription)
John Baxter born abt 1801 Kelton, Kirkcudbright – travelling tinsmith
Elizabeth Baxter, born abt 1799 Cathcart Renfrewshire – travelling tinsmith
Anne Baxter, born abt 1836 Mauchline Ayrshire ( I assume daughter of the above)
Alexander Baxter born abt 1857 Lanarkshire (there are a few prison records for him in the 1870s)
Joseph Baxter born abt 1860 Berwick on Tweed.
I think this may be the family in 1841 – lodging in tents
https://www.freecen.org.uk/search_records/5a1421f3f4040b9d6e0bb2ba/john-baxter-1841-ayrshire-ochiltree-1800-?locale=enThe wife and mother here may be Ann Watson - at least according birth cert for Alexander, b 1857.
There are 1861 Dumfries prison records for John Baxter ( a tramp and tinsmith) born 1800 in Kelton, Kirkcudbrightshire. And again in 1864, with same birth information.
Regular prison records for him in 1846, 1849, 1850, 1851 & 1859 - all have born in Kelton, 1800-1803