I’ll always have two tubes, glue and patches, an extensive tool kit. After over 60 years riding road bikes, I’m still amazed at how many riders are so ill prepared. The hot dog Ricky racers on their $5,000 plastic bikes are more concerned about Weight than being prepared.
I used to get flats so often that carrying two tubes and a floor pump along with me is standard procedure. I might rethink now that I have Schwalbe Marathon Plus tires. I got a pair last year and have not had flat yet. They also show little wear after about 3000 miles. I think the cheap tires I used to use were about worn out with similar mileage. The Schwalbe Marathon Plus tires are very much worth their high price in my opinion.
I once got 4 puctures in one day (one nail and 3 thorns). Then I switched to schwalbe marathon mondial tyres but carry a spare slime tube and a puncture repair kit on my bike. Also I change the tubes and tyres and tubes every 2 years.
Wet tyres do seem to cut more easily, but there's not much you can do if you live somewhere like PNW. Adding some air may help, as sharp stuff can ping off the tyre rather than dig into the tread, and try to stick to the part of the road or path that other vehicles have been clearing with their tyres - not easy on a trike. Carry spare tubes, because patching is more difficult in the rain, and try to find the cause of the puncture and remove it from the tyre. Or trust to tubeless and carry a plug kit (but still carry tubes).
Tubeless doesn't warrant a mention? You forgot to tell us how airless tyres rattle your brains and have poor grip, how slow and heavy Marathon Plus is, how hard it is to fix a slimed tube. My advice would be to learn how to fix punctures, it's not hard and you won't have to do it often (unless you're in goathead thorn country, in which case all bets are off).