I think 10-12 applications is generally reasonable, though I also agree that admissions and cost are very unpredictable, and you could conceivably get into some of those very competitive and wealthy and meet-full-need schools, which would argue for taking a shot at a few extra of those, especially if you can get fee waivers.
So one strategy:
Pick a half dozen of those really competitive schools, that also meet full need (you can google colleges that meet full financial need), and take a shot. Pay attention to fit for YOU. For example, University of Pennsylvania (Penn) is right in Philadelphia and feels very urban. Dartmouth is in a very small, bucolic town in New Hampshire, surrounded by dairy farms and mountains. Dartmouth would appeal more to me. How about for you? Both Ivy League, but very different places to live for 4 years. Look at a Fiske guide or something similar to narrow that down. For example, read the entries for the Ivies, if you really want to go to one of those, and see if a few seem to jibe better with your personality, etc.
Limit your “target” or “match” group to 3 or 4 schools that you think might really suit you. Think about what you think might fit: urban (think Boston U.) v. college town (think Dartmouth); large (20,000+ students) v. small LAC (2000); warm v. cold (lots of great schools are in New England and Midwest and can be very cold); big-time sports v. D3. With your stats, I think VERY GENERALLY schools with 20-40% acceptance rates would fall into this category, though that would certainly vary and I only mean that as the most general of guidelines. One school with an 18% RD acceptance rate might look like a solid bet and another a long shot, based on geography, gender, financial aid policies, applicant pools, etc. For your general stats, I’m thinking schools like Holy Cross, Boston College, Wake Forest, Wisconsin, Georgia, Case Western, Rhodes College–I mean in terms of admissions being more or less in that range. These are very different schools and aid would be more or less likely at different ones.
Pick maybe 2-3 schools that look like sure bets in terms of admission AND have an affordable price, probably some of your in-state schools.
Using this strategy, you would apply to 11-13 and have a shot at a half dozen of the really competitive schools. (Why are they so competitive? Supply and demand, not you or your academic potential. I think the numbers at one of the Ivies last year was something like 40,000 applications and 2,500 acceptances. Think about if you entered a lottery with 40,000 tickets, with 2500 winners, and you had one ticket. Way more than 2,500 of those applicants are exceptionally qualified and would succeed at the school. The school simply can’t accommodate all the great students who want to go there.)
But anyway, you could use the schools’ net price calculators (NPC) to eliminate some schools. For example, if you had a list of 15, ran the NPCs on those, and 3 came out high, you could eliminate them to get down to a dozen.
I agree with the comment above to work on those applications hard over the break. It’s better to be in early, and some schools do have early financial aid deadlines. Some have already passed. Good luck!