While I don't have the 18th century lineage of St. Lawrence Adams, I can fill in a few of your blanks. I am only recently coming up to speed on this. My mother is the one with the records. We are direct descendants of S.L.A and I offered to see what I could find on the web. We are related through one of two sons by his wife, LUCINDA. My mother lists the son as Timothy Herbert Adams (1799-1867), married to Susanna Mervine. This would seem to be the same person you list as "Herbert T." There is a Mervine Street in Philadelphia (no proof it is the same family). The second son, David, is buried in Hollywood Cemetery in Richmond, Virginia. Hollywood has a David Adams on their on-line records buried there in 1865. I found no record of St. Lawrence, but my mother's notes say he died in Virginia in 1832, so your two S.L.A.s ARE likely to be the same man. Timothy was buried in Monument Cemetery near Temple University in Philadelphia. Monument was dismantled at some point and my mother has records of where his grave was moved.
My lineage by generation:
St. Lawrence Adams (m. Lucinda) Timothy Herbert Adams Francis Warner Adams (1832-1916) William Herbert Adams (m. Mary Harry Short) Mary Harry (Harrison) Adams Aman (m. Ralph Lincoln Aman) Alice Adams Aman Pitt (m. G. Selden Pitt) Myself
Let me know if you are interested in any of these generations. I won't bother with it now since you may be going another direction in your search.
My mother's records on St. Lawrence include immigration records from 1793 listing location as Great Britain and Ireland. He was what Americans call a Scotch Irish Presbyterian, many of whom came from Ulster, but we have no proof of his Irish birthplace, but I haven't really looked yet sine I don't know where to start looking. He was naturalized in America in 1793. My mother has copies of census records showing Philadelphia addresses. He was a shoe maker and appeared to have lived above his shop at one point. He was a member of the 1st Presbyterian Church on what is now Market St (then High St.). My mother has seen the church record when she visited the church in the 1970's of the baptisms of sons Timothy and David, along with wife, Lucinda! She made notes, but does not have a copy. There was no record of his marriage to Lucinda. They may have married in Ireland before they moved.
Hope some of this helps. I would be very interested if you manage to find the Irish connection, and to find out what your connection is to the family.