Hi Caubeen,
Sorry to confuse you with my Ulsterisms! You are absolutely right, in Ulster Irish both a dh and an mh (and a bh on occasion, in words like banbh, garbh, tarbh which are pronounced bannoo, garroo, tarroo) are pronounced as an -oo-. So the Ulster Irish for think is smaoin(t)eamh, which is pronounced smween(ch)oo. A hair cut is a bearradh gruaige which is barroo groo-igya. Déanamh (to do or make) is pronounced jannoo in Ulster.
So Seosamh is pronounced "Shaw-soo" and in the genitive (in phrases like Joseph's car - carr Sheosaimh) it is Sheosaimh (hyaw-siv), with the v sound pronounced.
Speakers of other dialects tend to find Ulster Irish too soft, with too much oo-ing, and rather flat in intonation, but to us many speakers of southern dialects sound very up and down in intonation and very harsh in the way sounds are pronounced - like a Dalek singing. (No offence, guys!)
So to recap, Solamh is solloo in the north, solluv in the south . In general I am not fond of Biblical names, but Solamh is one I like.