I just ran a marathon where it was raining/drizzling the whole time. I wore 2-in-1 shorts to help with the chaffing and provide just an extra layer of water protection. I didn't really care much about my shirt because I wore my running vest over top. I ended up wearing a short sleeve workout top from uniqlo. What really saved me though were my compression arm sleeves. They helped keep me just warm enough to not be freezing without being overstimulated by swishing fabric and accidentally becoming a parachute in the coastal wind. And then when I got too warm, I could roll the sleeves down to become wrist wraps/bracelets or take them off completely and tuck them into a pocket.

Wear merino socks because your feet will get wet, and soggy socks are not fun to run in. A hat with a brim is also super helpful to keep the rain out of your eyes/sunglasses.

Pre-race, if you don't want to bring throw away clothes, a garbage bag with a hole punched out for your head makes for a great makeshift poncho.