Monica,
Yes, the Lesmahagow records are worth a look, there is a list of mortcloths from 1765 that ends at 1819, then a list of deaths with varying information in sometimes quite a difficult script, from 1820 - 1844.
Other good Lanarkshire parishes are -
Avondale (621) Has a mortality book 1817 - February 1854. Good information, wife usually has her maiden surname, some place of abode given, age, relationship, place of burial, date of burial and occupation.
Cambuslang (627) has burials 1784 - November 1854, quite good and gets better in the last decades.
Carluke (629) has a very good list of burials in the churchyard 1838 - 1851.
Carstairs (633) deaths 1819 - 1854. Gives name, age and cause of death.
Crawfordjohn (636) deaths 1817 - October 1854 with good information, age, relationships, abode, dates of death and burial.
Douglas (641) Deaths 1833- 1854 gives name, age, abode, sometimes relationship, cause of death, dates.
East Kilbride (643) deaths 1821 - 1839, name, age, place of birth, cause of death, date of death, some dates of burial.
Glassford (645) Mortality book October 1783 - December 1807.
Govan (644) burials in Govan churchyard January 1817 - February 1819, also April 1820 - February 1855.
Libberton and Quothquan (650) burials 1783 - 1854.
New Monkland (651) has good mortcloths scattered throughout most of the books, in session accounts etc.
Pettinain (653) deaths 1829 - 1849 good condition, gives name, age, abode, date of death and a few have relationships including mother's maiden surname, but not many like that.
Rutherglen (654) Mortality book 1705 - 1794 good condition, gives relationships.
Shotts (655) various deaths, burials and mortcloths, some very good.
Wandell and Lamington (659) deaths 1820 - 1856 good information - name, age, spouse, maiden surname, cause of death and place of burial.
And, as usual, the ones you want are missing!