Thanks for the advice all, especially @wintertree. My experience went as follows:
Started off from a car park opposite Seaham Hall and headed south on the beach towards Seaham in the driving rain. This bit was nice enough but I wish I'd started from Nose's Point because the walk "through" Seaham was pretty bad, and I don't mean the weather - trudging along the A812 passing the Argos, Asda and anonymous warehouses for best part of a mile topped off by some boxes that turned out to be council buildings. Headed down the slag slope to Blast Beach which was great and the rain had stopped (for good) by now - despite weather forecast of 100% certainty of rain most of the day.
Explored the arches to the left before walking to the end of the beach, eschewing the steps 2/3 the way along and scrambling up the grassy slope to the footpath, made the mistake of looking back and seeing the council buildings scarring the landscape again!
There was nice little wooded section, on what seemed to be a mostly barren coastline, by the train line and some good steps down to the beach and that was virtually my last experience of the coast path as such, since I stuck to the shore from now on since luckily the tide was out all day. Mixture of pebbly/rocky beaches and just rocky sections, one of which may only have been passable an hour or so either way of low tide. Enjoyable although the almost permanent presences of slag shelves at the back of most beaches was mildly depressing.
I had left my water in the car and didn't realise at the time that Seaham Asda was my last chance to buy some, so had my fill from the brook * at Fox Hole Dene. Plenty of oystercatchers, some other shorebirds and a couple of cormorants, but virtually no humans spotted after Blast Beach. From guesswork, I don't think the actual coast path was particularly splendid, better on the beaches.
That said, even in the sunshine, I did become aware of how tedious it suddenly seemed, no more birds, a huge slag shelf running along an unvarying dirty coloured beach .... just as wintertree had suggested, and luckily I found my self at Castle Eden, so following his sage advice headed inland under the impressive viaduct, through the dark, long culvert and enjoyed a lovely autumn woodland walk. I missed Gunner's Pool Bridge though and emerged at St James Church.
* 36 hours later some serious chunder action started, very foolish.