Does anyone have experience in dealing with an elderly relative who lives alone, with no family in the same state?

No, it does not require multi hour emergency trips.

You call an ambulance and send it over there. Also, you don’t tell her you are calling one. You just send one over.

It's hard to reason with someone with dementia. Quite often, they don't "see" the reality that they need help.

Sadly, you may need to stop showing up for emergencies. In an "emergency" you call ,911 and have the helpers assist her, transporting her to the ER if necessary. Call the hospital and tell them she lives alone and there is no one nearby to assist.

The social services folks at the hospital will do a better job of pointing out reality to her than folks who are "just" her relatives.

With my own mom, she called with "emergencies" 3 days running, causing me to have an to leave work, drive over an hour to tend to her.

On Day 3 I sat her down and said "Mom, I can't do this anymore ". She really had NO idea that what she was doing was an inconvenience. I said I would no longer be showing up. She moved, with great reluctance into AL the following week.

She clearly isn't making rational choices, so chances are she is in the beginnings of dementia. Part of what happens to people with dementia is that they lose their ability to empathize with others' situations... this is why she doesn't care that you're breaking your necks making "emergency trips" over and over just to orbit around her and give the appearance that she's "independent" in her own home.

If no one is her PoA then contact APS and report her as a vulnerable adult. Or, if she requires a trip to the ER, tell the discharge nurse that she is an "unsafe discharge". This may be the opportunity to transition her directly to a facility near you.

By doing the "emergency trips" you are basically enabling the ability to deny that there must be a higher level of care. The entire family should get together to make it HONESTLY clear to the elder that this isn't working; short of that it is APS and asking for wellness checks. If there is competency and no POA then this remains in the elder's hands. There WILL be an incident; whether it results in care or in injury and placement, it will come. Not that I mean that to be a comfort, but some elders wish to die at home and would rather gamble on going fast, and feet first out the door.

Keep trying. That's about all you can do. You might encourage a visit to you, and some visits to places you have pre-examined. The lure of being near you and frequent visits may eventually help.