The last scheduled steam-hauled service ran into Liverpool Exchange on 3 August, 1968. But it was manned by a Lostock Hall (Preston) crew.
The last steam special left Liverpool Lime Str on Sunday, 11 August - the (in)famous 15 Geunia special. I don't know if it had an Edge Hill crew or not. As it went to Carlisle, I suspect it would more likely be a Manchester or Carnforth crew, with an Inspector from Liverpool who knew the road.
The Liverpool district lost its last steam turns earlier in '68, not sure when, probably around April.
IIRC, the last two sheds with steam were Edge Hill (8A) and Speke Junction (8C, I think it was). Not sure when Birkenhead went, probably the same time.
There were many more sheds previously, eg Bank Hall and Aintree - I guess these were Lancashire & Yorkshire Rly sheds back in the day. So it's an enormous subject, both in time and locality. Far better if you can give more information to narrow down your search.