UTIs, diabetes and more I’m sure.Has anyone been in this position?

There doe not appear anyond dropped the ball. ICU care means acute illness.
I think you need some perspective...
Day 1 infection starts
Day 3 swings in sugars. Suspect, UTI? Doctor or PA orders a sample.
Day 4 sample picked up for the lab, then there is a 3 day wait
Day 7 results and antibiotic ordered. From here the order comes from a local pharmacy to be delivered to the NH. Do not expect drugs to sit in a NH pharmacy ready to go.

It looks like the interception and trip to the ER happened early in the process. Did aunt use Lantus or sliding scale insulin because if the med was not in her med cart, there would have been delays to order it.
Now sliding scale insulin orders are within a set of parameters. Give too much can cause life threatening low sugars. Her ordered dose was probably given at the maximum and sugars were still high, then she needed more care. The right call was made.
Diabetics are vulnerable to infections. You are looking at sugar swings which is a symptom of the UTI. Keep in mind that lab results take 3 days to ensure she is getting the targeting antibiotic.
I suspect she may have been going into septic shock or there was concern of it, which kills. Monitoring with insulin was probably needed in ICU and this kind of care was well beyond the NH

Firstly falls are very very common in hospitals. Now that we don't bind people up when they are confused the falls are even MORE common. Sad but true. People enter ill. Sometimes confused. Sometimes combative. They do not, as you can imagine, have a caregiver right at their side at all times. That would mean a hospital with 250 patients has 250 caregivers, and that's not the case. So it happens, especially happening before full assessment and understanding mobility and mentation. So there is that.
I don't really understand your relation to aunt, but it looks, if you got the medicaid work done that you may be POA? Or is the cousin the POA and the aunt her mother? If you fill out your profile we can get a better picture.
As Aunt is in hospital it is the POA for health care involved now if aunt is confused. And yes, decisions need to be made, especially as involves whether aunt is a full code blue or not, if she wants full resuscitation were she to die, or not. Intubation and ventilator or not.
If the diabetes is this out of control her management may be beyond your average nursing home and she may need to go into SNF (skilled nursing facility) for some time to try to stablize the sugars.
I am so sorry. You have been through a lot with aunt and sounds like more is still getting piled on the plate. Sure do wish you good luck and hope you'll update us.

Thanks for your responses. I think we are more concerned about how the sugar levels got so out of control at the NH and how do you let a patient fall off the gurney at the ER My cousin has a call into the NH to speak to the director of nursing and we don’t want to upset anyone but someone seems to have dropped the ball on this journey

I hope everything goes well for you regarding getting your aunt approved for Medicaid.

Keep us posted on how your aunt is doing.

Let the NH and hospital work out the transfer. This is out of your hands, Infections do cause wild sugar swings. They were probably worried for a seizure. When Aunt is discharged, there may be medicine changes. Medicaid will be involved and not you.
Try to relax and be less anxious. You can visit her with less physical activity,rather than care for her. Keep the visits positive.