I'm prepared to come over in September, and I'm planning on just staying a year before I go back to school to complete my Master's degree. I would love to be in Busan or Seoul, but I'm open to other areas. I've been working part-time as a teacher/tutor here in the states and I've realized I work much better with smaller groups. I would love a school where classes are between 6-10 kids or even 1-on-1 tutoring.

It is extremely easy to get an entry-level position at almost any Hakwon, an English Academy, which is basically a franchised or independently owned for-profit extracurricular English teaching/ tutoring center. I guess they are comparable to a sylvan learning center or a Kumon Math and Reading Center, but you'll be doing more teaching instead of tutoring each student one-on-one.
The good part about this kind of work is that you'll get what you want in terms of class size and the exact/preferred location/ city that you want to be in as with Hakwons you'll be directly applying and interviewing with the school. You can easily look for some Hakwon positions by going to a website like https://www.eslrok.com/en/teach-in-korea.

However, please keep in mind that working in Korea is not like working in the States or other highly developed countries with professional workplace standards.
There are a lot of Hakwon horror stories out there that date as far back as the early-mid 1990s and a lot of these issues are still there in 2022.
So, here are some things to keep in mind, the starting salaries are incredibly low here, in 2022 you will be offered a starting salary of most likely 2.1  or 2.2, which was offered to teachers back in 2002 when the cost of living was 50 ~ 70% lower than it is today. Since you have experience teaching in the states, try to push for a higher 2.4 or 2.5 starting wage as much as you can. Also, I am assuming that you are not an overseas Korean? If you are, then you can hold an F-4 visa, which gives you complete financial freedom, but if you don't have Korean heritage you'll have to apply for the E-2 (language Teaching Visa), which is sponsored by your employer and means you become tied down to the school that sponsors your visa for the duration of that 1 year. It will be almost impossible to switch jobs in the case you hate it at the Hakwon, are overworked and underpaid, treated badly, screamed at, and belittled (which happens often), so please make sure to request to talk to a teacher that is currently working there and ask a lot of questions including:

How many foreign teachers that are working there have re-signed their contracts and how long have they continued to work there?
(If teachers have stayed on for a 2nd, 3rth, 4th year and beyond, it's a good sign).

How many teaching hours are there per week? (Usually, 25 ~ 30 teaching hours is a good amount for a Hakwon) which is like 5 ~ 6 classes per day.

Is there a curriculum in place for the teachers to follow? And how is the lesson planning? Is it minimal or is there a lot of things that the teachers have to create, plan, and design themselves?
A good Hakwon should have a curriculum in place for the teachers to follow. Some are disorganized and expect the teachers to make everything :<

Is there technology in the classrooms? If the classrooms are equipped with a TV then you can use PowerPoint, the textbook CDs, and other visuals to help students learn.

Do you get your own lunch break, which you don't have to spend eating and helping the students. (Required by law to have an 1-hour break per 8 hour work day, or 30 min per every 4 hours worked).