You’re asking a lot of the right questions. I would suggest filling out the Net Price Calculator (NPC) at Vanderbilt, Wash U., and Williams. If those schools don’t get into the $20-25k budget, then you can ignore all schools that don’t offer merit aid.
Have you thought about UL-Lafayette as an in-state safety? Your stats would get you their top merit aid package, which would essentially be a full ride (tuition, room & board, and an on-campus job if you want it…research a likely possibility). It has less than 14k undergrads, compared to the 31k of Arkansas.
Also, familiarity can breed contempt, but I would also suggest at least touring LSU and its Honors College before eliminating it. Ditto with Tulane, as I think you’d stand a very decent chance for a full tuition scholarship coming from Louisiana. (Are you in the New Orleans area? If so, then there are even more options for $ at Tulane.)
If you like Arkansas, you may want to check out Hendrix which has a flagship match program.
If you’re a National Merit Semifinalist and like mountains, check out West Virginia University. I think you would get at least free tuition there. This post talks about where to find those scholarship details, and that whole thread is very informative about the school in general. There are about 20k undergrads at WVU, so about 1/3 smaller than at Arkansas.
I’d take a look at Washington & Lee. It’s an academically strong school near the mountains in Virginia. Only about 9% of students without need receive merit aid, but the average merit aid package to those students is over $48k (source). Although there is a significant portion of students who participate in Greek life, because it’s such a high proportion, I have been told that Greek life is not at all like the stereotypical one. @DramaMama2021 or @cinnamon1212 might be able to shed more light on this, if W&L is of interest to you.
Two other academically strong schools that you might want to investigate are Furman and Wofford, both in South Carolina. You may also want to take a look at Roanoke, UNC-Asheville (North Carolina’s public liberal arts school), and Appalachian State. I suspect you’d be very competitive for their most generous merit aid offers. A bit further from the mountains, but not too far, is Davidson in North Carolina, another strong academic institution that offers generous merit aid to the approximately 6% of students without need who get some (average of over $40k)
A definite reach for sufficient merit aid would be at Colorado College. This is another academically strong school with very little merit aid, making it a big reach (due to difficulty of getting sufficient merit). There is a small handful of full tuition scholarships for STEM students that you can apply for, though (source).
Looking for big, easy merit aid, some other schools you may want to consider if you get National Merit Semi-Finalist status are: U. of Louisville (16k undergrads), U. of Maine (9k undergrads), and perhaps others. If you’re a finalist, take a look at U. of Kentucky (22k undergrads), Alabama (32k undergrads), and others.
Arizona has dry mountains rather than the verdant green of the Appalachians, but if they appeal, check out U. of Arizona (well-reputed honors college and very generous merit aid, but a big school), U. of New Mexico (15k undergrads), and/or New Mexico State (11k undergrads). U. of Wyoming only has about 9k undergrads, and you’d get into budget for sure based on its merit table (source).