All you can do is apply. Yes, LACs offer excellent aid but as I showed above, their acceptance rates for international are tiny. Even the ones that claim to meet need and are need blind - have low rates. You can try of course, but this will be your issue.
You mentioned Bama. It is cheap for international relative to others but not cheap to meet your $20K. @Andygp if he’s still out there and his son has been at Wooster a few years but can point you to some of the cheapest out there.
I know on another thread, someone said their intl friend got a full ride at Southern Miss so not sure the normal international merit and Texas A&I International plus Minot State might be others to look at. But $20K, I’m not sure.
You might find this thread a bit disheartening and frustrating but it likely has good info for you to source additional names.