Are hospitals discharging senior patients to out-of-state nursing care facilities?

IMO, no they would not send them out of State. Not because of Medicare, that goes from State to State, but because of Medicaid which doesn't. If there is any chance that Medicaid will be involved in her care, she can't be sent out of State. A Hospital does not have that kind of control when it comes to placing someone into Rehab or a NH. What happens is their discharge person calls around to Facilities near you to see if they have a bed. Then they tell the family what beds are available. The family then choses which facilty they want their LO to go to. If the family can prove they can care for the person at home the patient can be released to them. Never think tbat Hospitals, Rehabs and NHs have that kind of control. These places are not prisons.

It's very typical for elders to show a decline after getting discharged from the hospital. Some even have delirium for a few weeks afterward. Had she been in need of rehab during THIS hospitalization, the doctor would have written an order for it. If she's very out of sorts a month after coming home, I'd alert her PCP that something doesn't seem right, and she needs a physical exam (along with a cognition test). Definitely check her medication list, as suggested, to make sure there's nothing on there that shouldn't be, or perhaps a new med order was written in the hospital that you are unaware of. Meds often have side effects, so if that's the case, call the PCP again and find out if this med is necessary or contributing to her state of mind.

For whatever reason you are asking such an odd question, I have no idea is the answer. I had to admit both of my parents 2x apiece into rehab and had a choice of where I wanted them to go. So it would be unthinkable that NO facility would be available in the entire state of Oregon, forcing the hospital to consider out of state rehab.

If/when she's hospitalized again, ask that very question to the Social Services people.

We know no details of your mom's diagnosis or prognosis, but I think your connecting the hospitalization with the decline may be circumstantial. That's just a guess.
I think that it would be a rare thing that there would be no SNF or Rehab in your area, and really can only guess as to what hospitals do about discharge to rehab if you are bordering on other states. It is a good question to ask when she is hospitalized and the Social Services department would be the ones to tell you. I sure do wish you good luck.