Maybe in some locations/subjects. My stare pays pretty well. The nicer areas have no trouble getting teachers and it can be difficult to get in there (same for private schools).
What I have seen and been told is that the places that have the worst time hiring have some combination of safety concerns, students that don't care and misbehave, and oppressive bureaucracy or policies that they don't agree with.
So in summary, there's no shortage in my opinion. There's competition to get a job in a good school. The bad schools can't seem to hire people because people don't want to work in a shithole. Same sort of thing is happening in other jobs with the great resignation/reshuffling when managers ask why they can't find candidates.
I don't believe America is running out of teachers. You can argue pay is bad for K-12 but from the high school teachers I've talked to it seems to be the under-high school crowd that bears the worst of it.
Some of this has to do with the average skill set required. While teachers should be paid more, in America the barrier to entry for teaching is so low under the collegiate level that it's almost an utter triviality to become one. As a result, many well intentioned but less than capable people become them. My experience going through K-12 in America was primarily that school is simply a day care for most people. If university didn't prepare me for the real world K-12 definitely didn't prepare me for the real world. I don't believe paying teachers more would ameliorate this. This is simply a function of capability of the teachers. Compound this with lower pay due to lower barrier to entry and you have a reasonably high turnover. Much like most government jobs, teaching below the university level is a passion field where pay is low and motivation is uncharacteristically high.
A lot of schools require teachers to pay for their own supplies as well. This typically is where the good teachers, the passionate and educated ones, leave. You'd have to be a fool to make $20-30/hr. and end up paying 1/5 out for books, materials, etc to enhance your kids learning.
America does not invest in education like it used to. K-12 has been an indoctrination camp for children for a while. If you doubt this, ask why home economics was removed from virtually all schools, and why there is a loud bell in the first place (hint: it's to condition you to old school lunch/break bells). Or, why American schools in general are built to feel like prisons. After grade school I never remember getting to see out of classes because they were all concrete jungles with high walls and thick metal gates to keep you in. At my schools (early 00's) there were also actual armed police everywhere and monitors patrolling the halls. I got the hint by high school that schools aren't set up for learning, they're set up to condition you to authority. In hindsight it was a claustrophobic experience.
So if teachers that can teach can't afford it, and people who want to teach aren't capable, and America itself has designed public education to be more of a prison indoctrination camp than mind opening, why would anyone stay?
Most people take about two years to get a teaching certificate but about half of them never teach because they discover they hate the job.
One problem is that teaching certificates are not usually portable between states. A friend of mine got a certificate in NY and moved to NV to teach, although it went badly when the school district disapproved of his politics, his apartment burned down, and he got beaten up by a cop. NY on the other hand would not accept a teaching certificate from another state.