To clarify, are the Annan records not available online or are they totally non existent? Would be happy to go to Annan if I could see them there.
There is almost never any point in going on spec to the place where early records were generated, because next to none of them will still be held there.
The Church of Scotland registers of baptism, marriage and burials are held by the Registrar General for Scotland in New Register House in Edinburgh, and the contents have been digitised and indexed and can be viewed on Scotland's People (SP). You can see a detailed list of what registers have survived at
https://www.nrscotland.gov.uk/research/guides/old-parish-registers/list-of-old-parish-registersMany registers of dissenting and free churches are held in the National Records of Scotland (NRS). Most of these have been digitised and are available on SP under 'Other churches'. You can search to see if a particular congregation's records have survived but not been digitised by going to
https://catalogue.nrscotland.gov.uk/nrsonlinecatalogue/search.aspx, typing the name of the congregation in the 'Search for' box and 'CH3' in the 'Reference' box.
The Church of Scotland's Kirk Session (KS) records are almost all in the NRS, and many of them have been digitised and can be viewed on SP. To find out which Church of Scotland KS records have survived, including those that have yet to be made available online, go to
https://catalogue.nrscotland.gov.uk/nrsonlinecatalogue/search.aspx, type 'Kirk Session' in the 'Search for' box and 'CH2' in the reference box. If you want to know whether the KS records of a specific parish have survived, add the parish name in the 'Search for' box.
The same search, using CH3 instead of CH2, will produce a list of KS records of dissenting churches held in the NRS.
The Roman Catholic records are not held by NRS, but they have been digitised and are available on SP.
If SP and NRS don't have the records you are looking for, or if you are looking for Episcopalian, Methodist, Congregational, Jewish, Quaker or other minor denominations, try the Scottish Archive Network
https://www.scan.org.uk/ to see if they are held in a local or university archive.
Local archives should know if there are any surviving documents held elsewhere. For Annan, start with
https://www.dumgal.gov.uk/article/15308/Local-archives