Like others, I am so sorry to hear about your baby's bottom with the rashes and pain. I know it hurts you as much as her. I hope it improves.
If it were me, I would begin by changing your detergent. There is a special detergent for babies' clothing that doesn't have any of the fragrances, dyes, or chemicals in it. I'm allergic to Tide as are our girls. I only noticed the girls were allergic to it after my MIL washed some clothes at her house in Tide and they began having the mad itch. I had to tell my MIL again that if she washed their clothes to NOT use Tide.
As they got older, I was able to switch over to Gain or the detergent sold at Sears in the buckets for the HE machines b/c we have a matching washer/dryer frontloader HE machines w/ the add-on risers, which lifts the machines about 13-inches from the floor and the risers have a drawer in them so I can put my bottle of Woolite for my undergarments that can't be washed in anything but Woolite (my HE3T that came from Sears has a hand-wash cycle so I no longer have to hand-wash my matching undergarments (cami tops, bras and matching underwear).
Then, if it were me, I would have the carpet taken out along with the padding beneath, sweep and/or vacuum the flooring beneath and have fresh padding and carpet put in. You can usually get a pretty good deal from carpet installers who lay carpets for huge houses and they have all these left over pieces that will fit perfectly in rooms of middle-income families with houses that are anywhere from 1100 to 1600 sq ft (our house is 1600 sq ft but we have linoleum in the laundry room, kitchen, both bathrooms, and the spa room)
My thoughts are that it is something in the carpet or the padding that is not coming out with vacuuming or steam cleaning the carpet. A lot of crud gets caught in the padding and unless you have a high-end vacuum cleaner you're not going to get whatever is in that carpet out and sometimes even with a high-end vacuum and steam cleaner you're not going to get whatever is in that padding out.
If replacing the carpet is not an option, then considering having a professional steam cleaning service to come out to clean the carpet and let them know ahead of time about the baby and how you are not sure but there could be something in the carpeting that is causing the problems. It wouldn't hurt to have them clean any cloth furniture you have while they are there. Often these companies will give you a discount on cleaning the furniture if you have them professionally clean the carpets.
I would also make an appointment with a highly recommended dermatologist for your child. If ANYONE can get to the bottom, no pun intended, of the baby's tushie rash's origin, it will be a dermatologist as they are trained to recognize and/or send lab tests off to find the underlying cause of the rash then how to proceed with treatment.
The last thing I can recommend is allowing the baby's bottom to get as much air as possible. If you have a playpen for the baby, lay the baby in the playpen without a diaper so the bottom can get some air and help it begin to heal.
Often, rashes on baby's bottoms tend to worsen over time because the rash is covered in a diaper that allows any moisture - not necessary a wet diaper from pee or stool but moisture from sweating, is the perfect breeding ground for bacteria. Let your child spend some time in their "birthday suit" for a little while each day. It may produce greater results than you might think.
The reason I believe it may be exacerbated at this point is the fact that there has been no true underlying cause for the rash diagnosed so of course, the medication is not working. Then there is the fact that you moved from an area of the country that has a much lower humidity to an area where the humidity is extremely high. I can relate because I've lived all over the south an lower south east during my life. The further south you go, the worse humidity will get. Ohio temperatures may rise but the humidity is nowhere near as high as say what it is here where I live in NC (100 miles inland from the ocean but still in a high-humid area; my children both had to go al naturale in order to clear up a diaper rash faster b/c no matter how frequently I changed diapers, it was hot and it was humid and the heat and humidity are generally about the same. It's sticky heat, if you know what I mean....
Plus, you are in a new home. There may be something in the house causing a problem and that problem could very well be the carpet or the padding beneath. if you have never had or never seen old carpeting pulled up before you would be amazed at the grime and grit that gets caught not only IN the padding but BENEATH IT where nothing short of changing the carpet and padding completely or possibly having a professional service come out though I am not 100% sure if they can even remove what is stuck under the padding and if not, eventually it will work it's way back up through the padding and into the carpet thus causing the same problems to return.
I hope you find something that helps. But please, do try to make an appointment with a dermatologist who can examine your child's bottom as well as taking swabs or scraping to send off to get the cultures tested so you can get to the root of the problem and start on the cure.
All my best wishes and hopes for a successful recovery for your baby.