What kind of accommodation do I need to ensure I keep my current job / position at work?

I think from your responses to us LindaM--and thank you for them--you are not seeing your needs as very severe at all. Natural remedies for anxiety is indicative of (to me) just your everyday "that's life" sort of thing. We all have it.
I can't honestly tell, but either you have a condition that precludes your doing what is expected or you do not.
If you do, it needs to be documented by an expert.
If you don't you won't get cut much slack.
I just mentioned below that my daughter, 62 and nearing her plan to retire, does teacher aid work with the kindergarten crowd. She is expected to be in their tiny little chairs or on the floor with them. At this point that is pretty excruciating for her, but she gets cut NO SLACK whatsoever despite recommendations by letter from her MD that she not be up and down off sitting on floor with toddler types. None. The school system's attitude is "either you can do the job or you can't". We are, of course, talking a school system absolutely overrun with medical and behavioral problems that have staff resigning in droves. But they just hire new ones who last long as they last and on it all goes.

If your employer have an ER (Employee Relations) you will need to contact them, they will help you navigate the process.
There are multiple options generally speaking and it is all dependent on the company policies.

As you mentioned there is:

1: Family and Medical Leave Act (FMLA) - where eligible employees can take up to 12 weeks of unpaid, job-protected leave per year for specified family and medical reasons, including caring for a parent with a serious health condition.

2: Flexible Work Arrangements where employers may offer flexible work arrangements such as telecommuting, flexible hours, or part-time work to accommodate caregiving responsibilities.

3: Sick Leave or Paid Time Off (PTO) - where some employers provide paid sick leave or PTO that can be used for caregiving purposes.

4: Disability Accommodations - where if you as the caregiver or their parent has a disability covered by the Americans with Disabilities Act (ADA), reasonable accommodations may be required, such as telework arrangements or modified schedules.

5: State and Local Laws - There are some states or localities that have additional leave laws or caregiving accommodations that may apply.

6: Employer Policies - Reviewing the employer's specific policies on caregiving, remote work, and leave can provide additional insight into available exemptions and accommodations.

It is very important for you to communicate openly and honestly with your employer or employee relations or HR about your situation and explore available options for balancing work and caregiving responsibilities. Employers may be willing to work with you as the caregiver to find solutions that meet both the your needs and the needs of the company.

Best wishes

Hi Linda - I think you've gotten really great advice from others already - in fact, I was going to suggest the same as Alva's feedback - but you responded that submitting a doctor's note regarding anxiety was too severe. But, wondering what other options you'd have since you stated that now "your anxiety meter is way up."

You also referenced that "you're overwhelmed with so much going on." - can you elaborate on what additional is going on? Maybe if you explained that, it would help to provide other suggestions. Are you currently caregiving for your mother? Would this new role impede this in any way? Does the new role require additional hours worked or any conflict regarding whatever you referenced as "all that's going on with you?" Do you want to explain that to your supervisor and that taking on a new role at this time would be difficult based on that?

You mentioned that training in another department requires meeting a "rate" or you get written up and / or fired - but that happens in any position if the employee's performance isn't up to proper standards - even the position that you currently have. Do you think that you're worrying more than need be because it's the fear of learning something new and having a current comfort level?

If you need to request something from your FLMA team, what are you requesting that you need - time off to take care of your mother? Or that due to stress from caregiving for your elderly mother, it would not be an appropriate time to focus on a new role? Can you just tell the FLMA team that? Perhaps your primary doctor can provide a note. I'm a bit confused - Maybe if you can kindly provide more info that would help!

I must say it sounds kind of fishy, after 10 yrs of working and they want to place you in a job where its sort of pass or fail? In my experience, you may be entitled to something by law and employers have to abide by it, like FMLA. But when time comes they feel they want to get rid of an employee, they will do it. Putting them in a job where they may fail is one of them.

Happened to my grandson. He has epilepsey so when his Warehouse had COVID his employer had to allow employees to take time off. My GS did. Before this, he had been put on a job he just could not do because of his ADHD. He requested to go back to his old job, and it was allowed to. He got time off and took it in days and hours. I told him he should take it as weeks. Taking days and hrs meant they needed to find someone to do his job that day. When he came back from the COVID break, they put him back on the job he could not do. He went to HR to see about Disbility for upcoming surgery. At the end of the day, he was called into HR and fired basically for not being able to do his job. He then told HR because of his ADHD he was not able to do that particular job. They still fired him. Equal opportunity was able to get him a small settlement but it took a while. He worked there 5 years and never had a write up or warnings. He was given promotions because he did his jobs well. But I think even though he was entitled to the times he took off, it was not OK with those over him. He was able to find another job and has done well there.

I have worked many places and I can tell you when they are trying to get rid of someone. Be careful.

Hi there. I have depression and anxiety. Hereditary. Suffered most of my life until I got help in my late 20s. My work life is pretty normal. I'm an introvert, but I have to deal with people at work, which can get stressful. I work the usual 8a-5p schedule.

Know why I can do all that? MEDS. Very moderate doses thereof. I see my psychiatrist every 3 months to monitor how I'm doing and if I need dosages adjusted. I haven't needed adjustment for awhile now. I take them only as prescribed. 

First off: NATURAL HERBS DO NOT WORK. Not as good as meds, anyway. Even the tone of your post sounds anxious (not meant as an insult, just saying that's how it comes across) and are immediately saying no to meds that will actually work. And if you're already spiraling with the "I just know I'll get fired!", then you need legit help. Herbs are like putting a band-aid on a deep cut. Helps a little maybe, but doesn't stop the bleeding. "Natural" is not a code word for "safer than meds". They aren't. A med has to go through years of testing before it is approved to be prescribed. Look at the bottle of whatever herb you're taking, and you will see a disclaimer that it is not meant to treat or prevent any medical condition. "Natural" items are required by law to have that disclaimer. 

You say you don't want psych meds on your work record. No need to worry about that, because of the magic that is HIPAA. Your meds or med history can't be disclosed to your employer. Yes, even if your insurance is through your employer. 

You might be afraid of meds, and I get that. I promise you meds aren't just for people in the streets who mumble to themselves. No psych doctor wants their patients to be zombified. I still get anxious, but nowhere near as severely as I used to. It doesn't swallow me whole anymore. Anxiety is not a fun way to live. It is a chemical condition and you need meds for that, not something that grows in someone's backyard. You don't have to feel like this and there is relief out there.

Have you considered meds for your anxiety? If not, why not?

Is your issue inability to focus? Or is it anxiety?

See your doctor right away.
You can get anxiety or stress (or both) disorder written up after interview with psychologist referral. They will know how to do this as they DO do this.