Dealing with elderly parents is a huge stress for the children. I've seen that in my time volunteering at an Assisted Living facility.
Generally what I saw was that the couple tried living at home and after Dad died, it was a lot easier to convince Mom to move in to Assisted Living.
Let's ignore the other factors; they're important but not critical.
The most important thing to realize is that your parents right now are in a state of denial. They don't want to accept that their youth is leaving, never to return. It's a scary time for people. They obviously expect that you will take care of them, which is the norm in many cultures around the world.
I'd start by saying in clear English, "Your plan of my moving across the country is not going to work. I have no job here and cannot afford to live here, taking care of you, and not have a full-time job." There may be objections, but that's reality. Most adults don't have the strength to work 40 hours and then take care of ailing parents full-time on top of that. In addition, how would you take care of yourself? This makes an assumption that leads toward depression and loneliness for you.
I'd also quit nudging them. This needs to be brought out into the open and clearly. I'd say this: "We all know that I live across the country. We need to determine living situations for you and that can't include me. In an emergency, it would take at least a day for me to get here. What if you fell? Would you want to lay there for a full day for me to get there, assuming that you can somehow contact me?" If they say, "Don't worry", I'd add "Why not? I worry about you. How will you take care of this?"
If they say, "You will take care of us!", the answer then is "How? I am not a trained health care worker. I have no experience or expertise in elder care. I have no desire to give you both baths and am completely unqualified to manage medicines and monitor medical conditions for you. This is a job for someone with skills and experience, which I do not have." I've seen firsthand what's needed, and it is a lot - especially when the health and memory start to fail.
In the end, they may need to experience a hard lesson firsthand. They may need to experience something and realize that they need help, and the distance makes it impossible for you to be there quickly. It's only after they end their denial that this discussion can really happen in earnest.