Nigel Hardie has transcribed and published parish of Selkirk deaths for 1748-1778, 1819-1835 and 1836-1854.
Selkirkshire Antiquarian Society has published CDs containing transcripts and digital photographs of gravestones in the Auld Kirkyard, Brierylaw, and Shawfield Cemeteries.
An earlier publication by the Selkirkshire Antiquarian Society listing gravestones in Selkirk Auld Kirkyard is Gravestone inscriptions prior to 1855: Vol 1. Selkirk, Ashkirk and Lindean Old Churchyards published 1969 (listed in the LDS Family History Library catalogue in microfilm format, so hopefully available worldwide in LDS family history centres).
Note: Lindean cemetery lies within the old Galashiels parish.