There are a couple of issues here.
Drying -- a truly clean vessel should "draw clean" when rinsed and should dry fairly quickly. Your issue is that water is beading up on the inside. Your option is to use a lint-free towel to press down inside the vase and wipe it dry OR to clean the glass so that the water is naturally shed. You can consider using a salt & vinegar paste, a baking soda & vinegar solution inside, or even denture cleaning tablets. You may also need a bottle brush.
Cloudy -- this is due to mineral deposits in your water. Again a solution that will dissolve calcium, such as distilled vinegar, and a slight abrasive such as baking soda, salt, or a brush, should help you remove the build up.