As far as I could establish (as I can't walk too far myself), there are no "roads" to these West Coast beaches as the West Coast comprises of wall to wall hotels. However, all the beaches are public and there [b]should be [/b]signs showing the footpaths. There's Payne's Bay and Batts Rock. Nearer to Bridgetown is Brandon Beach which is nice if you just look straight ahead to avoid the port. It has a nice eaterie called Weisers where you can rent a brolly and chairs plus shaded parking under the trees and on the beach, also full facilities. If you need to get a car nearer the beach then there was a park at Surfside (Holetown) and also at the Folkstone Marine Park, a little further on, where you could also park in the shade, rent sun beds, with full facilities. Heading towards Port Charles you could also turn left just before Speightstown and find shaded parking on the beach but not sign posted.
You only mention the West Coast which is pretty eroded - why not try the East Coast - it's fantastic and there is parking. Try Consett Bay - south from Bathsheba, a grass triangle in the road has a sign post to the satellite station (usually a couple of cows grazing) follow it down and there is a lovely shaded bay - refreshments at weekends only so take your own.
Have a great time