You should not be able to see fluid, be it spit or condensation, coming out of the upper holes. I wonder whether the pads are closing correctly.
If in fact you feel air escaping from your lips (not going down the saxophone "tube"), then you have embouchure problems - might benefit from lessons and/or a softer reed.
Similarly, if screws are working ...
According to Charles Harris, who has been playing saxophone 35 years and been repairing them since 25 years, he has answered this question in a relative way on qoura.com
You can read the whole thing here:
Hope this helps!
As said in the comments, what you observe is likely to be condensation, which is a normal phenomenon on a sax.
However, it is a good habit to clean it with a pad saver after you played in order to remove the moisture from inside the sax, preventing corrosion and eventual pad damaging… After cleaning with the pad saver, do not let the saver inside the sax as ...