If you're looking at Whitby in particular, try Ingrid Flute holidays (can't remember the web address but it's easy to find via a search). We have used them a few times, over a period of 17 years (last time was last year), and they are very friendly, efficient and the cottages just as described. Very reasonable prices too - last year we had a gorgeous 4-bed townhouse near the river for only about 600 pounds for a week in August.
Ingrid Flute have cottages all around the North Yorkshire area, so good whether you want coastal, rural or small town.