I guess the next questions are: Is it a drainage issue, where water drains to the run and stays there? Or is it rain that got in because the run isn't covered? How big is the run?
If the water isn't standing in puddles, you can put just about anything down. I'd start with a bag of pine shavings to begin soaking up the moisture. Then add dry leaves, grass, a little straw - whatever you have on hand. Build it up as time passes into a nice, deep litter. But do cover your run if you can - at least on the side that the most rain seems to blow in.
At 6 weeks yours should be pretty well feathered, so you could start by turning off the heat lamp. You could even crack a window in the room the brooder is located in during they daytime. They'll scream bloody murder when they lose the light 24/7, because they hate change. But they'll get over it, I promise. Oh, and take them outside for regular jaunts if you can confine them. I start mine outside in a pen within the run we have. Works great for me, even here in northern Wyoming. I raise all my chicks out there, even when temps are in the teens and twenties, and have had no losses or issues whatsoever. One of the reasons I raise chicks the way I do is so that they learn from day one that nighttime is for sleeping, not for running around cheeping all night, or for eating around the clock. So they are very happy with going to bed as the sun goes down and waking up when the sun comes up. No adjustment to temperature fluctuations and they experience the natural day/night cycle. Yours haven't had that advantage, so it will take them a bit to get used to the change.
The one year I did raise chicks under a heat lamp in the house, I finally couldn't handle it. I evicted them to the coop before it was even finished! Temps were in the 20s and dropping through the night. They were 5.5 weeks old, and it was on April 1. I put the heat lamp out there for them and then spent the entire first night jumping out of bed and checking on them. They were fine, snuggled down into the corner by the pop door, nowhere near the heat lamp. The same story the second day, except I only checked on them once. The morning of the third day I took the heat lamp out - if they weren't going to use it I wasn't going to risk a coop fire. That night it snowed. And we kept getting snow until June 6th. They thrived.
So I decided right then and there that the conventional wisdom of 95 degrees the first week, 90 the second, and so on was pretty much hooey. After all, if you watch a broody hen with chicks hatched out during colder months, she doesn't have a thermostat under her wings to adjust the heat, nor does she have a night light under there for them. They run all around their environment and just duck under her for a quick warmup, or to sleep at night, or if they get spooked. They do this regardless of the outside temperatures. Sometimes it felt like I was supposed to keep chicks in the house until I found my first egg in the brooder. <sigh> Good luck!