That’s a tight budget for that area. As others said, there is no “just outside the city”. All of the cities run together.
I would suggest asking about “mare motels”. These are very open barns with larger stalls/runs. They give your horse the opportunity to walk around, socialize, and get some sunshine. Sometimes they are even cheaper than a box stall. But it’s not enough room to run or buck or play.
As someone else mentioned, pens or other outdoor board might be an option, and I would certainly consider that. There’s really no significant weather that a horse needs to be protected from.
I think you can find all day turnout, but you may need to increase your budget. But all day turnout is often solo in a sun pen. I do think the horses enjoy the change of scenery, but it’s not the “run and buck and play” turnout.
Because of the lack of turnout, some places offer AM lunging, hot walkers, arena turnout, etc. This is helpful for high energy horses (assuming you ride most evenings so they are getting some exercise 2x a day). It may also help your horse adjust to a more confined schedule.
It’s not all doom and gloom though. Horses are happy and healthy and live good lives. There are lots of boarding options, trainers, service providers, etc. But it’s not what you’re used to. You will need to compromise in big ways (usually either commute time, price, or services), and it will be a bit of an adjustment for you and your horse (mostly you).