My feet don't sweat much.
To each her own but as I said I have been attending F1 and other motorsports events multiple times per year since 1978. Most race days involve probably 5-7 miles of walking.
In the past 10 years or so I have switched from trainers to a nice supportive sandal like the Clark's Sea Breeze flip flop (or Brinkley which has a cute flower on the thong part), which are readily available in multiple colors on Amazon.
They are supportive enough for walking and obviously don't get soaked the way trainers would, on rainy days. No socks needed, just a cute pedicure. Plus if you get black, silver, bright pink or whatever, they can pair nicely with a dress for evening. I have lived in them for four or five days on long weekend breaks and never felt the need for alternative shoes.
(Again, keeping in mind this isn't home; at home of course I wouldn't wear the same shoes four days running. But I have come to loathe hauling large suitcases around. And it's all the same in the end; no one notices or cares.)
OP, are you attending as normal spectators or will you be hosted in team-related hospitality tents or suites? That would be a place of refuge if it's raining.
Wide-brim roll-up sun hats also are good for keeping the rain off your face. I never bother with waterproofs or ponchos any more, but they do sell plastic ponchos with hoods that come in a pocket-size pouch. There's a 12-pack of them on Amazon right now for 10 quid.