Me and my bud Dev detect not too far from Scarbro (closer to Whitby) We enjoy around 300 acres of mixed pasture and plough and have had finds from Henry 111 to Viccy sort of eras. Scarborough has a Castle overlooking one of its bays and being a fair sized oldish resort should def produce some beach finds if there is not too much sand present. The nearby Abbey to which you refer could well be Whitby abbey which is 7th century - there are other Abbeys and castles not too far off Reivaulx abbey and Helsmley castle are reasonably close by ancient standards...
Sure you will enjoy your time at Scarborough
Stu
