How to get esa letter when depression makes starting anything feel impossible

I know my dog helps. My psychiatrist has said as much. My lease is up in a few months and almost everything in my budget has pet restrictions so I need to sort out documentation but every time I try to start researching how to get an esa letter I end up overwhelmed and close the tab The phone calls, the figuring out which service is legitimate, the potential for a landlord to reject it anyway, it all feels like too much when I'm already running on empty. Has anyone been in this exact headspace and actually gotten through it? How hard is the process really when you're not at your best?