//I am surprised that there is any road where one can do 37 mph in Dorset.//
The best place where one can certainly get up to a good lick is the delightfully named "Puddletown Bypass" on the A35. This is just about the only stretch of that road that warrants its status as a "trunk" road. The stretch is about 7 or 8 miles long and bypasses not only Puddletown but also Turners Puddle, Briantspuddle, Affpuddle and Tolpuddle (where the martyrs come from). East of it there is a long stretch of single track road stretching to the outskirts of Poole (unless you take the A31 where there is a long stretch of 40mph limit through Corfe Mullen and the surrounds). To the west it becomes single track (albeit quite decent on the Dorchester Bypass) before reverting to "country lane" status up to Winterbourne Abbas. A couple of short dual carriageway stretches exist beyond there - just long enough to enable you to get past the caravans and tankers before you hit the congestion as the road approaches Bridport.
It is a well known spot for mobile speed cameras and one or two cases of grossly excessive speed (one, a Porsche, at 155mph) have been reported. But it seems a permanent solution may soon be found
https://www.wessexfm.com/news/dorset-news/2853376/plans-for-speed-cameras-on-the-a35-puddletown-bypass/