My Dad is 91 and the nursing facility told me the reason for his grogginess and constant sleepiness is due to his cortisone shot? Any ideas?

I agree with Funkygrandma59, cortisone is a steroid and shouldn't cause someone to feel sleepy. I've also had them and have not felt this aftereffect.

Are you his medical PoA or medical representative? Has he had a diagnosis of cognitive impairment by a doctor so that the medical PoA is now in force? If not, then this is probably the reason why he is going on his own to appointments and you are not being informed.

He could be having slurred speech due to some other medical/health incident such as overmedication, not taking prescribed medication, a stroke, etc. Your best pathway forward to help him is to find out who has the legal authority to accompany him to appointments and discuss his private medical information with his doctors, and making decisions in his best interest.

While shots of cortisone can cause sleepiness, it is much more common for them to cause SLEEPLESSNESS, and also some agitation, some agitation, headachy feeling, sometimes briefly increased urination. I am wondering what other medications he is getting and I feel the POA should get a full list of medications.
Can you tell me why Dad is getting shots of cortisone?

I doubt it is the cortisone shot that's making him sleepy, as I have had more of those than I can count and they have always given me insomnia, which is a side effect of them. It sounds more like they have him on some other types of medications that are keeping him in a drugged state. I would call his Dr and the nursing home Dr to find out exactly what medications they have him on and why. They may need to be adjusted or even stopped. Something sounds very fishy to me. Good luck getting to the bottom of this.