Chick very sick

Hi all! I'm new here and I'm so glad I've joined because it has helped me out so much!
I've had my chicks about 2 weeks now and yesterday I went to Tractor Supply to get fencing and they had some chicks that I thought were so cute so I got 2 austrolorps and 2 barred rock. So that was last night around 5 that I got home with them. Well 1 of the austrolorps was not looking so great in the middle of the night but I wasn't really sure what to do with it. I turned the heat lamp on thinking maybe since it was new it needed more heat than it was getting from the rest of the chicks, but this morning it looked dead. It was all wet and laying on its back with its wings sprawled out and the other chicks were pecking at it. It was still alive, just barely. I picked it up with a towel and it was covered in poop. It didn't put its wings back in, and its breathing was very labored and shallow. I separated it from the others, which I now realize I should have quarantined the new ones right from the beginning. I'm not sure what might be wrong with it, but now the other austrolorp isn't looking so great. It doesn't look "sick," but just isn't moving around much at all.
If anyone has any recommendations on what I should do or might know what's wrong with the chick, please let me know.
Thank you in advance.

What is the temp under the heat in the brooder? Baby chicks MUST have the right amount of heat, and most likely it got chilled, or perhaps got trampled in it's attempt to seek heat. Call the store, and see if they are loosing many of their BA chicks. YOu can give them Poultry Nutri Drench. How big is your brooder? How many chicks? At 2 weeks, they should have about 2 s.f./chick. You can get away with tighter spacing if you are brooding a LOT of chicks. But, for a small brood, 2 s.f./bird is a good size.

The temperature inside of my house is 70 and then with the heat lamp it's usually around 90 when it's on for a little while. There are 18 chicks in the little chick pen (probably only about 3x3ft) I have under the lamp. The chicks I had were already a week old when I got them, so they are about 3 weeks old now. The new ones were only a week old. I'm not sure if it was alright to mix them together? They told me it was fine. Please let me know if I'm doing anything wrong because I really would like to make sure all my little guys will be happy and healthy.

Your 3 week old chicks should be completely off heat now if they are still in the house. That brooder is only big enough for 9 chicks, and by the time they are 3 weeks old, they should have about 2 s.f./chick. The biggest issue is that your younger chicks still need a bit of heat, while the older ones should be weaning off it. By the time they are 4 - 5 weeks old, they should be completely off heat and moved into their outdoor coop. When you say that the temp with the heat lamp is "usually around 90 when it's on for a little while. " I'm wondering what it is if it's on all night? That 3 x 3 brooder is not big enough to give any kind of temp variation, and way over crowded. Could you pick up an appliance box? Perhaps a refrigerator box, and use that for a brooder? Or even move all the chicks into your coop so they will have some room to spread out and adjust their heat exposure by getting closer to or further away from the light?

I haven't left it on long really and certainly not on all night. It's in my bedroom because I have cats and dogs and a 6 and 1 year old that I don't want to get to it when I'm not looking lol. Hubby is laid off of work still right now so he has had them outside most of the day when it's been nice out, but it's going to be rainy here the next couple of days. Is it alright for them to be out in the coop when it's rainy and in the 50s? If so then I will just keep the older ones out there. Also, the light is about 4 feet over the pen that they are in and only over half of it. I have ducks in another pen so the lamp is over top of both of them in the middle. I don't really think it need to be on much because it is very warm in my house. I really just turn it on when they are huddling a lot and seem cold.