I would suggest either rinsing it in a bathroom somewhere, and/or carrying some sort of pouch or container you can stick it in so you don't dirty your pocket. Then you could clean it at home. In a pinch you could probably pour water from your water bottle on/through it outside if you carry a water bottle already. Could also wrap it in a handkerchief if you carry anything like that.

I don't normally carry any of that stuff to be honest. I don't usually even wear a coat. Just my phone, keys and sometimes a wallet.

I live in Spain so 80% of the year I don't use a coat. And water I would just buy when I need it :)

When I go hiking I do bring those things but then I don't come across places that sell iced drinks :)

That sounds like a lot more effort than just avoiding places that give you shitty paper straws instead of proper plastic ones