She cannot leave me alone for ten minutes; she follows me around or wants me to stay with her while she is in bed. She says she is dying. The doctor said everyone is dying. How can I have her tested for either dementia or Alzheimer's? as well as other psychological problems.
There could be other, more subtle things that are frightening her. Your profile states that she lives at home, but I have the impression from your post that she lives with you. Is your father living with you also?
If they are living alone, they're both at ages where they can recognize physical limitations; your mother might be afraid because her mobility has been compromised, or the same situation for your father. She may fear that she can't help him if something happens.
I can honestly state that I'm in my early 70s and I've become more concerned about safety because I know I'm not as limber as I used to was. And that "safety" includes safety from unknowns, such as strangers at the door (which I rarely answer), more concern for people who might wander into the yard while I'm gardening (I usually have a pitchfork with me), etc.
If your parents watch the news and see all the horrific reports that some stations milk for all they're worth, those kinds of events can cause an underlying fear that's ongoing. Although I have no scientific proof, I think a lot of people developed a high level of anxiety after watching the repeated videos of 9-11. I still feel that fear when some news stations drags out the videos and runs them again.
Have there been any notable crimes in her area?
Other than that, what are her medical conditions, specifically, does she have extensive mobility limitations, has either she or your father been diagnosed with a serious condition recently? If she's just begun using a walker or something that is mobility related, she could be sensing the loss of physical dexterity and become worried what she would do if she fell or if something else happened.
Or maybe she's just lonely. Do you have children, grandchildren, pets? If there are pets, bring a cat or dog to her and let her pet it while you're trying to work. Or put on some of her favorite music. Hopefully it will calm her so that she doesn't feel the need to be so close to you.
If she won't agree to an evaluation, I wouldn't push it; just deal with the situation as if she had been evaluated for dementia - the diagnosis would confirm or refute any suspicion of dementia, and you could get meds if that's the plan, but the issue of the day to day interaction would still be one to face and address.
I just searched this site for posts or articles on "Dementia Clinginess"; there are a lot of hits on this topic. Perhaps there are some suggestions in other posts that might be of help:
https://www.agingcare.com/search.aspx?searchterm=dementia+clinginess