@SmiYa0126 Thanks so much!! :') I'm so sorry about your pullets. Chickens are such interesting and funny little individuals, it doesn't take long to get attached and break your heart when they go! It is encouraging to hear you've had greater success raising your own.
@Eggcessive , thank you! I *think* what I ordered comes as a loose powder, so I'll have to figure out how to measure it out. For now though - I was able to get a refill on the liquid solution from my vet's office today even though she wasn't in. Despite the weirdness around the other Dr.'s offer to euth, I love this office, they always do their best.

I sent my vet an email around the same time I posted here, and she responded this morning:
"I doubt coccidia is the cause but it is not an impossibility. There are 3 systems that void into the cloaca: renal, reproductive, and digestive. My suspicion is that it is reproductive (cancer, infection, trauma or even something like hyperplasia) but cannot rule out the other systems without diagnostics. Your dose for SMZTMP is correct at 1.6mL by mouth twice daily for at least 7 days. It shouldn't hurt her and may help." (I told her I was prepared to euthanize with CO2 if Sprinkle takes a turn for the worst.) "I would take it one day at a time. As long as the bleeding is not worsening and she is going about normal chicken business, then giving her time is fine. If she is hanging in there an exam may be helpful to see if there is something that can be done."
It sounds like there are a number of potential causes, some of them terminal, but for now I'll see how Sprink responds to the sulfatrim. She's walking, vocalizing, resisting handling, but she doesn't seem to be feeling great. I was out running errands today and I think she was hiding out in a nest box all day. When I brought her in, she produced this heinous-smelling liquid dropping (looks/smells like it contains old brown blood, hopefully that means no active bleed). She wasn't interested in eating scrambled egg. I suspect her crop is slightly sour -- it was squishy and her breath was a little off. I gave her some miconazole and tube-fed 20 mL of electrolyte water. I'm worried about her being weakened by dehydration, between the diarrhea and blood loss. I think I'll keep her inside tonight and, assuming I can get her crop moving, try to push fluids aggressively to see if she perks up again.
Wish us luck!
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