Please don’t think I’m trying to be a naysayer, but you need to aim for colleges that are not top tier. Your GPA and test scores are probably going to rule out top colleges, barring some hook. You should not “still” be aiming for top schools, because in all likelihood, you are going to be throwing away money on apps. Sure, go for a high reach, if your family can afford it, but the majority of your list should comprise schools that fit your budget and where you have a good chance of being accepted. Aim for colleges where your GPA and test scores put you above the 50th percentile of accepted students, and ideally, above the 75th.
Your budget means that you most likely should stick to public universities, or aim for colleges where you might get some merit aid. Ideally, you should take the SAT again and try to get your score above 1400 for a better chance of being offered money. Your GPA is good, but it will not get your application a second look at any top tier colleges. Your SAT score will be low for top colleges.
As far as asking for recs, yes, you need to ask in person. You’re going to college in just over a year. You will have to speak with professors and manage yourself in an adult fashion. Shy people aren’t given a pass on advocating for themselves. My daughter is very shy and learned in her first year of college that not talking with professors is a mistake. It means you won’t be on their radar for any recommendations or opportunities that arise. You will need to seek out your professors as soon as you realize you are not clear on the material. So start being proactive now. Simply asking the teacher for a rec will go a long way in helping a teacher get to know you.