No joy tracking down the elusive William, I'm afraid - but following the son, John, it looks to me like the family were members of the Free Church, which will make finding any trace of them in the parish records before 1855 a bit tricky, as Scotland's People only indexes pre-1855 parish records from the established Church of Scotland.
I'm guessing you already know this, but for the benefit of others, son John Murray (abt. 1820) married Janet Turnbull (abt. 1820, daughter of Charles Turnbull and Margaret Thomson) sometime before 1844 - no marriage record that I can find. They had at least 10 children between 1844 and 1863. Family Search indexes some Free Church records, and John and Janet's five youngest children were baptised in the Free Church in Airdrie.
Janet Turnbull died on 26 Nov 1882, and John married again on 27 Jan 1891, to Jeanie McArthur (abt. 1843, daughter of John McArthur and Margaret Forgil (sic)), in Stirling, after banns according to the Free Church. He died a year later on 27 Jan 1901.
I can't find John Murray or Janet Turnbull in 1841, although I do have Janet Turnbull's parents at St Ninian's, Stirlingshire, with two of Janet's siblings.
As for father William and mother Annie Welsh - can't find any trace of either of them before William shows up in 1861 in Maryhill with John and Janet, although if I was a betting person, I'd say Dod likely has the correct banns proclamation from Cadder in 1814

Ruth