I have emailed Carnwath district library to see if they have any old parish registers for a marriage between John and Jean, but if you haven't found anything, it may suggest the document doesn't exist.
All the surviving registers of the Church of Scotland were collected in 1855 by the Registrar General for Scotland, and they form the basis of the Scotland's People database. Other churches' registers are either still with the respective churches or in the National Records of Scotland or a few in local archives. The Roman Catholic ones and many of the Secession (Free) Churches' registers are available through Scotland's People. There is a published list of the extant Secession Churches' registers, but it contains no listings of pre-1855 registers from Carnwath.
Interesting to see "Bothea"s surname is TOWERS (ie her maiden name) and not husband William's surname of RAE. I have heard of Scots unofficially 'marrying' using the handfasting ceremony...so I wonder if this couple did that. No parish records show a marriage for William RAE and Bethia TOWERS.
Is 'Bothea' from a transcription, or is it your own reading of the original? In older documents 'e' and 'o' can look very similar. I wouldn't read anything into it.
As for her surname, it's not unusual for married women, and quite common for widows, to be listed in the census and other documents by their maiden surnames. In Scotland, a married woman does not legally lose her own surname, and in legal documents she is usually named as xxx yyy or zzz, where xxx is her given names(s), yyy is her maiden name and zzz is her husband's surname. This is why you usually get a mother's maiden surname in Scottish baptism records (if the mother is named at all!).
Here's an extract from the Rothes parish register:
"1748, September 11th. William lawful son to Alexander Leslie of Balnageith in Burncrooks and Mrs Anne Duff his spouse was baptized" , and here's another, this one from Brechin:
"1834, 26th June. James Hay merchant in Arbroath and Margaret Sang, widow, residing in Timber Market in this parish were contracted in order to marriage and having been regularly proclaimed were married the 30th June". Margaret died in 1870, and her death certificate confirms that Sang was her maiden surname and she was married twice, first to Andrew Watt and second to James Hay.
If the baptisms of the children of William Rae and Bethia Towers say they were 'lawful' (or some contraction of 'lawful') that means that the parish clerk was satisfied that they were born in wedlock. As long as they don't say 'natural child' or 'begotten in fornication' or something of the sort it is reasonable to suppose that the clerk believed that the parents were legally married.
As for handfasting, if a couple publicly promised to marry one another and then produced a child, that was enough to constitute a legally valid marriage, even if the Kirk frowned upon such marriages. See
https://www.gla.ac.uk/schools/socialpolitical/research/economicsocialhistory/historymedicine/scottishwayofbirthanddeath/marriage/