My club Martinstown CC is 15-20 minutes drive from Weymouth. One of the prettiest grounds in the county and certainly the best track. If you want to PM then I can put you in touch with our Fixtures Sec.
Weymouth themselves and Dorchester may host you, Portland Red Triangle (although not the nicest ground to play at) would be worth trying. Then there are other village clubs such as Charlton Down, Cerne Valley, Bere Regis and Cattistock (try Liam SCCC on here) who are a bit further away but also worth a go.
Weymouth is the main night out around here, unless you want to go as far as Bournemouth. Not sure where is best to stay but a number of our lads are from Weymouth so may be able to advise further.