Should I bother applying to graduate school?

I’m a fourth year undergrad and have been planning on applying to graduate school, but I’m now having second thoughts because I’m not sure I even have a chance at getting in. I would love to be in a TESOL (teaching English to speakers of other languages) program, as it’s a field I’m very passionate about. I’m an English literature major with a minor in linguistics.
My GPA and GRE scores are good, and I expect my recommendations to be pretty good (though I’m not totally confident that they will be outstanding, since I could have been a more active student in certain ways).
My other concern is that I don’t have enough experience to be considered. I only recently got a few months of experience as an ESL volunteer tutor. I had an internship in online education, and I am now participating in an internship that relates to both ESL and online education.
I would like to attend grad school right after undergrad, but now I’m wondering if I’d be wasting my time applying before I acquire more relevant experience.
I very much appreciate any insight on this. Thank you so much.

Maybe I also should have mentioned that I would apply mostly to public schools.

Well, the only way to tell, really, is to apply. You can potentially apply to 2-3 top-choice programs and see what happens, and if you don’t get admitted then go back, get some work experience and try with a bigger round in a few years.