Ian, difficult to answer your question as I don't know your exact route but I go from Poole to Colchester regularly and use A12, M25[Dartford Bridge],M3,M27,A31.
I don't know if Clacket Lane is too soon for you and this can be busy even on a Sunday. The next services are Fleet on the M3 which may be busy but I can recommend Winchester services which is one of the smaller ones and generally there is no problem parking with a caravan. Rownham services on the M27 are ok but more lorries to contend with but Sunday should be no problem. Once on the A31 through the New Forest there are several lay byes seperated from the carriageway. There is not much past Ringwood but there are places just of the main road.
Have a great time in Dorset, I hope the weather improves it is dismal tonight in Wimborne.
If you have any particular questions or an idea where you may take a break then please ask.
Ray