I've not done St Oswalds in its entirety but I visit the coastal strip of Northumberland quite regularly. Which end are you starting from?
The first (or last) couple of miles take you along the Pilgrims Causeway which crosses the sands from Holy Island to Beal on the mainland.you need to check the tide times and plan your day accordingly because the causeway is innundated by the tide for roughly two hours before and three hours after high water. The majority is on firm sand but it's muddy in places and very wet so you might find it easier to sling your boots round your neck and walk it barefoot. It's rather eerie walking out there once you get away from the roadway.
After an inland diversion from Beal, you'll be following the coast as far as Warwkworth. There are quite a few campsites on this section. If you click this link to UKCS Search it should list most of them on a map. The C&CC run two of them - Dunstan Hill and Beadnell Bay.
However, if you prefer to avoid campsites, there are quite a few places along the coast where you will get away with rough camping (I prefer to say 'rough camping' rather than 'wild camping' because the Northumberland coast is definitely not wilderness). Camping is not allowed but I've found if you pitch late and leave early in secluded spots you are unlikely to be moved on. Obviously avoid areas with easy road access. For instance, north of Embleton, the coast is mostly dunes and the coast road is never far away; south of Craster there are low cliffs and it is slightly more secluded.
From Warkworth the route follow the River Coquet (pronounced locally to rhyme with coke not cock) valley inland to Rothbury but I don't know of any campsites immediately around Rothbury itself however.
Good luck - enjoy the hike.
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