Not to minimize the possible seriousness of bloody diarrhea, however it is possible a bad reaction to a food change, in this case to adult food, could possibly cause this.
While any diarrhea is dangerous because of dehydration and underlying problems, we had one of our hospital cats do this a while back. We had been feeding her the same food we feed our patients and she was fine with that. However we got a ton of sample packets of a different food and figured we'd feed it to her. We did the food change gradually, but soon she started losing weight, losing her hair, getting scabs all over her body and had horrible bloody diarrhea. We couldn't find anything else wrong with her and changed her food back. After two days her stools firmed up, and after a week no new scabs. After two weeks you could see the hair trying to grow back.
I'm not saying that's what this particular cats problem is, however being it was mentioned she recently started eating a new food, it could be a problem. The cat still needs to see the vet though and soon to prevent further problems. Until you can get to the vet or at least a fecal sample, try to keep your cat in it's own food bowl, not the one that has the other food in it. If you have to feed the cats separately, it could make a big difference.
Is your vet always that busy? If so then I'd concider getting a second vet for emergencies or outright change vets. It's not good to have to wait 3 or more WEEKS when you have a problem now. Most vets should be able to get you in within a day or two, sooner if it's an emergency.
Good luck and keep us posted.
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