If you don't want to take buses then private driver is pretty much the only other option. I can't see you finding a tour that will cover those destinations.
As for Uda Walawe, jeep safari with a guide is the only option. You won't get into the park any other way. Your accommodation there should be able to organise a safari for you. To keep the cost down, you might be able to organise a group to go together. There will be lots of jeeps at or near the park gates if you prefer to negotiate your own deal. The park is well worth it whichever option you go for.