Calling her several times a day is unrealistic. For your own MH cut it down to once daily maximum. Ideally fading out to a few times a week, unless there's a period of particular concern about her health when you'd naturally check in more often until things improve. You sound stressed out and you need a break from being her listening ear.
There's numerous charities running helplines in UK which she could call for a listening ear. No it's not as good when the person you're speaking to doesn't know your history etc but you can't actually do anything to improve her life, so the fact the helpline people probably can't either makes no difference.
Is she doing all she can to look after herself? Anyone with problems needs to develop ways to relax, calm down and soothe their own soul that don't involve other people. It's not fair or realistic to rely on moaning to others and seeking sympathy, reassurance or validation from them constantly, to make oneself feel better.
Is she involved with MH services? Does she have a MH social worker or psychiatric nurse? If not, does she need to visit her GP to ask for a referral? For those whose lives are out of control and perhaps need a medication review or course of therapy etc these services can be a useful intervention to get things back on track.
Does she need elderly care? GP can refer to that social services department too, for a care needs assessment.
GP may also have information on charities running support groups or self-help groups or just social groups for isolated people. The people attending these things are often unwell to some extent and may be struggling. She doesn't have to be feeling fine to attend. She doesn't have to rely solely on you for companionship.
As well as whatever else is going on in the wider community, if she's well enough for ordinary hobbies. Community hall notice board, local library or local council website, as well as local free newspapers etc can be good sources of information for this type of thing.
If she's in the UK there's little help unless things are dire, so it could be true that she feels she needs more help, ideally. But if she's constantly voicing this it could be manipulation to guilt trip you into moving closer to her. Don't fall for it, you have your own life to lead, you're not responsible for a parent.
IMO it's unkind and unfair to keep telling someone who cares about you about your problems which they can't solve in any way, it's obvious that as someone who cares it'll stress them out. Sometimes people who are unwell can get so wrapped up in their own problems they don't stop to think about others feelings and needs. There's also a difference between keeping someone updated and mentioning how you feel about something versus moaning on about it frequently and at length.
A lot gets bandied around about it being good to talk, especially about MH struggles - but in all honesty, there's a limit to that. Sometimes no amount of talking will change anything and it might be better to focus on practical measures to improve life in some way or just making the best of things. People also have their own limitations on their capacity to listen. You're not obligated to be her emotional dumping ground for getting out all of her negativity. It's unreasonable to use another person in this way, except for the occasional need to vent.
You've offered her to move closer to you so you can provide more care and she's turned it down. That's her choice. It doesn't mean you have to think up other solutions to her problems, that's for her to do.
I think you sound far too enmeshed and need to step away and separate yourself from your mother.
Who are you?
What are your needs, goals and dreams?
What are you doing to work towards those?
How are you managing your own stress levels?
What changes would you like to make to your life?
You and your life matters too