I found the 1963 NER PTT on Timetable World, though I found it hard to navigate. The pattern of 2 out and backs from each end for Liverpool/Newcastle applied then, as in 1974, but the routes were more complicated.
The two morning trains from Newcastle were 8.42 and 9.49 am, the former via Darlington and York, the latter via Stockton and Ripon. The return workings were 3.0 and 5.5 pm from Liverpool, the former via Ripon and the latter via York, but both via Darlington.
The two morning trains from Liverpool were 9.0 and 11.0 am. I can't be sure of the route for the former it seems to run non-stop from Leeds to Darlington, the latter went via Ripon then Darlington. The return workings were 3.16 and 4.42 pm from Newcastle, both via Darlington but then the former via Ripon and the latter via York.
Unless I've missed something, nothing from Liverpool to Newcastle went via Stockton.
Back in the late 'sixties/early 'seventies, the mid-morning Newcastle-Liverpool ran via the Durham coast line, calling at Sunderland, West Hartlepool and departing from Stockton at 11 06 - diagrammed for a Gateshead 46. For the gaggle of spotters hanging around the Marlborough Road footbridge at Hartburn Junction, it was the highlight of our day....especially on the occasions when a 40 or 45 substituted for the booked 46. There was presumably a corresponding return working in the evening, although I don't recall ever having seen it. This was my default spotting location during the school holidays when staying with my grandparents on Teesside.
This does appear in Table 1 of my ER PTTs. The second morning Newcastle-Liverpool ran via Stockton until May 1972, the 1106 departure time at Stockton lasting to the bitter end!
But, as in 1963, there appears to be no return working via Stockton.
And to stray a bit off topic, do you remember the Newcastle-Colchester (summer)/Lincoln(winter) at Stockton? Bizarrely, that also had an unbalanced route with the Colchester (summer)/Lincoln(winter)-Newcastle going via Darlington.
The railway clearly expected more people to leave Stockton than to arrive!