Why pasta keeps always damp and moist when I cook it?
Refrigeration does not improve pasta texture; on that front my advice is to cook only as much as you are going to eat each time.

Other than that, it sounds like you are overcooking the pasta, so "when you judge it done" appears to be in error, and you should alter your judgement to an earlier point if you find the pasta too wet/soft they way you have been doing it. The time needed varies depending on the shape. When it is done it should be removed from the water (or the water removed from it) immediately.

I find no benefit to adding oil to the water - save the oil for adding just after it's been drained if you want it to do something useful.