My first tip. Practice in the rain. Though it might not apply to this situation now, think through the past year and think of the times you had available to play a quick round of dg, but decided not to because the weather was crappy. If you had gone out instead of watch netflix, you'd be mentally and physically prepared for this USDGC qualifying. Or from now on you'll be prepared. (Same goes for wind/snow/cold/dark - all the times that people are like "ehhh I'd rather not") And really, it's not as bad as you imagine it will be if you're prepared for it well.
Umbrella. I may have been at one point 'too cool' or maybe it was just a laziness thing, but seriously, if you look at the radar before the round, and it looks like rain, an umbrella is key. A big one. The biggest you can find (probably search umbrella on these forums, or browse the internets for one with good reviews, or go to the local golf store, they carry umbrellas for the purpose of playing golf in the rain.
Towels. A couple extra in baggies. I just take an old beach towel and rip it into strips 5" wide or so, cut the strips in half and shove each one into a zip lock bag. A dry towel ever few holes.
Chamois, leather, after using the towels, a leather chamois can add the finishing touch to make you confident in that grip.
Keep your throwing hand dry. Should be obvious, but do all handling of the bag, moving objects off the ground, touching anything that is wet with your off hand. The throwing hand is sacred, the only thing it should touch is a dry disc and the inside of your rain coat.
Rain coat. If you find a decent one that retains mobility, let me know. I have a $400 rain coat, it keeps me dry when it's raining, but damn am I restricted in that thing. I prefer to throw without it unless it's really coming down.
Boots. Ok, most people like to be light on their feet, so, waterproof shoes. But seriously, if it's coming down cats and dogs, everything is wet. If I have a well oiled piece of solid leather on my feet, I am so much more comfortable than a lightweight sopping wet (so not so light any more!) pair of athletic shoes.
Pants. There's probably some good options for lightweight athletic pants that will get wet, but not too uncomfortable. Like say jeans or heavy work pants that get wet and heavy. Or a pair of rain pants that could keep you dry and comfortable. I don't like rain pants if it's warm, but in cooler weather, that's the route I go.
Socks. I don't do the waterproof socks thing, but maybe that's a good answer to the wet shoes thing? I wear smartwool socks, or a medium weight hiking sock and have multiple pairs to change into throughout the day.
Plastic. I don't find it to matter much if I'm prepared and keeping discs dry, though DX type tends to retain a better grip when wet, I don't change my entire bag over.