Guns are made to be used, the finer the gun, the greater the enjoyment. I use all my guns, even the most valuable and tailor the usage to the type of hunt, not the weather. As mentioned, some of the finest, most valuable guns in the world get used regularly and frequently in Britain's foul climate. The normal thing to do there is to return your gun to the maker or a trusted gunsmith at the end of the season where it will be stripped, cleaned, polished, lubricated,adjusted and any repairs or parts replacement attended to. This has kept many 100-130 year old guns working and looking great. This also was why detachable locks as found on best guns by Westley Richards, Purdey, Holland and Holland and Lang became popular, to allow drying and lubrication at the end of a wet day.
Personally after a rare wet day ( I live in the desert now) I dry my gun with paper towel or rags before heading home, then at home it gets a thorough deep cleaning and relubrication. I find a hair drier ( don't tell my wife) very handy for warming and drying metal parts. Reexamination in 2-3 days. Been doing this for over 60 years while living in various Canadian climates, never had a prooblem.