I really urge you not to put yourself in the position of needing to “find the time and confidence to ask” your teachers for recs in the fall. Follow @privatebanker 's instructions and send the emails now. If you procrastinate, it will just weigh on you, and it will get harder and harder as you realize that the procrastination has put you in a position of being one of the annoying students with the last-minute recommendation requests. (Can you tell that I speak from experience?) Don’t be that student - commit to the requests by email now, so that you only have to follow up when you get back to school.
This is where some coaching could really help you - having someone to hold you accountable for goals like this, and even more importantly to use goals like this as a learning laboratory to deconstruct the thought processes that hold you back from tasks like these.
One best-of-both-worlds option for you might be the Cook Honors College at IUP. This would give you the advantages of a selective, small-college environment within a large public U that provides a wide variety of majors, at a public U price. Looking at their description of a “typical” student in their program, your GPA is a little below that mark but your SAT is above. Note that a teacher recommendation is part of the application. https://www.iup.edu/admissions/honors/typical-student/ The questions on both the student application and the teacher rec show that they’re looking for thoughtful, imaginative students, which I’m betting describes you well!
Clark is also a match for your stats - your GPA and SAT are both slightly above median - but boosting your test scores would be a help in securing enough merit to make it affordable. Did you do any test prep before getting that 1370? That’s not a bad score, at all, but if you got that without prep, then spending some time with the Khan Academy app this summer and retaking in the fall could open up options for you - either more competitive public U’s, or more merit at the private colleges that are already a match but might not give you enough merit money to hit your price-point without the score-boost.
Check out the book and website “Colleges that Change Lives” https://ctcl.org/category/college-profiles/ - there are quite a few schools there that could be good fits and have enough merit potential to be affordable.
Another option within your budget would be smaller public liberal arts colleges in other states. Look at UNC Asheville, St. Mary’s College of Maryland, Truman State in MO, and U of Minnesota Morris. These all offer both education majors (or at least minors) and social sciences.
I also think that it would be really really good to get your feet wet in terms of working with kids, before you go to college… both to build your confidence, and so that you’re not choosing a major path just based on hypotheticals. (You don’t really want student teaching to be your first real-world experience!) Maybe a program like City Year, where you could work with kids in school settings, would balance out your classroom experiences in high school with real-world experience that could bring what you want to do in college into focus, and help you to bring out the not-shy-with-kids person that you know you can be. At least consider getting some work experience with kids - whether in a beginning cheer program or just in a daycare or afterschool program.