Here's what I do when I need to go to a Bürgeramt:
For every type of appointment you can make, there should be an official website containing links to each Bürgeramt's calendar page. Bookmark that website, not the individual pages.
Find one or a few locations you'd prefer, then open each calendar in a separate browser! Opening multiple in one browser doesn't work, as it remembers your last selection per browser session.
Next, keep refreshing and checking the calendars every 30 minutes. Slots free up pretty often, but they're also full again soon after. If you're lucky, you can figure out when canceled appointments are entered into the system for your location (for mine it was 10AM every day). Around that time, there's a good chance you might even get a few slots that are only a week or two into the future.
Once you figured out when slots open up, check around that time daily. Book the first slot you can get, then keep doing this for a few days and book any slot that is better than your previous one (but be nice and cancel the old slot).
It's a lot of work, and it shouldn't be necessary, but I'd never go into a Bürgeramt without an appointment and this method has worked for me every time so far.