Westbury: Manors, A History of the County of Wiltshire: Volume 8: Warminister, Westbury and Whorwellsdown Hundreds (1965), pp. 148-63 (www.british-histroy.ac.uk).[Great site; search for John Adlam.]In referring to the manor at Westbury Leigh, this source states as follows:
"In 1545 the manor, with its captial messuage, and the lands leased with it to John Whatley, Leigh Common, and all appurtenances were conveyed to John Adlam, clothier, of Westbury....John died seised of the manor in the same year leaving his heirs his daughter Edith, wife of John Lambe, and Alice, wife of Robert Cogswell. ... The other half...passed on the death of Alice Cogswell in 1606 to her grandson Roger...."
Go to the website for the complete text and additional source information.
I think this is pretty clearly our Alice/Alicia.Though it seems commonly accepted that she died in 1603, not 1606 (source?), it is possible that she died in 1603 and the property did not pass to Roger until 1606.(It's unclear whether "in 1606" modifies "passed" or "on the death of Alice Cogswell.")I'm sure they even had delays in probate back then.Robert Cogswell and his wife did have a grandson named Roger, son of their oldest son, who was the only grandchild to receive a bequest under his father's will.So it all seems to fit.Hope this helps.Now we need to find out who John Adlam's wife was (some say Marjorie, but I haven't seen a source yet) and who their parents are.