I recommend that you swab out the upper tubing after each playing session to get rid of the excess water. What I do is use a cleaning rod wrapped with a strip of cloth to carefully swab out the upper slide tube. Always be careful of the lead pipe (“venturi”) in the upper tube.
All of our horns have removable crooks that can serve as an additional tuning slide. Because the chemicals in saliva can slowly leach the zinc out of the brass leaving pure copper behind (which will leave red spots commonly called “brass rot”), I rinse the crook with clear tap water. It takes about 20 seconds to rinse out the crook but over the next 20 years will be well worth it. Every other flush or so, leave the crook in place and fill the valve section with clear tap water (I use a mouthpiece as the funnel) and depress the 3rd valve so that the clear water also flows into the third valve slide. That way, the valve slide will be flushed and kept clean. It might take a couple of minutes but again, well worth the time.
When I just flush the crook, I also put some clear tap water on a “snake” and push it up the lower tube to dilute any saliva that collects there (obviated with the 3rd valve slide flush option).
Every year or so, I think it’s a good idea to have the horn professionally cleaned. (We call it a “chem flush” around here).