Since you mentioned Davidson I assume you will consider colleges in the south.
Davidson does offer full ride, full tuition and other institutional merit. It is highly competitive but worth researching.
URichmond and W&L have competitive merit.
In addition to those already mentioned upthread, other LACs that offer merit with acceptance (not competitive) to attract strong students: Furman, Rhodes. Both of those came in under $40k for my D.
For the best chance of merit, apply early. And for competitive merit, pay close attention to the deadlines as most are due by December 1 even if you are applying RD.