We’re in a very similar situation, and my daughter (singer - mezzo-soprano) is a senior currently playing the waiting game to find out who is the most generous with $$. We are also playing the numbers game and trying to get as much scholarship $$ as possible because while we are comfortable, we absolutely cannot pay full freight, and we are also sending another child to college in a couple of years, so we will save wherever we can. My daughter is also doing as many competitions as possible in order to collect some awards to cover whatever school scholarships don’t. Like you, we didn’t always have a lot to save for school, so while we have a small nest egg, it won’t come anywhere near what we need. And we absolutely don’t want her going into debt for a vocal performance degree.
My daughter has been accepted at UNT (it really is a great school, but we live in Texas, and she wants to go farther away, so she applied there as a “safety”), IU, and CMU, and we are still waiting on Oberlin, Eastman, CIM, and Juilliard. She has gotten three scholarships from UNT so far - 1 academic and 2 music. They are as generous as they can be, but they have less $$ to give than some of the conservatories. When she had a trial lesson with one of the voice teachers, he told her “we may not be able to compete with some of the other schools $$ wise,” which surprised me because most of the other schools at full price have $80-90k price tags. Right now, the UNT scholarships are covering about 80% of her tuition, which is great, but we’re hoping for at least full tuition somewhere (and would not turn down a full ride if offered!) 
She was also awarded scholarships at CIM ($25k), Oberlin ($25k), and Eastman ($15k) through outside competitions, but they were clear that “this is the MINIMUM you will receive,” so we don’t know what the price tag will end up being once all the #s come in. From what I’ve heard, Oberlin and CIM can both be very generous if they like you. Eastman is a little less known for its generosity, but we have high hopes. She auditioned for Juilliard mostly because she wants to be able to say she got in (it’s probably about 3rd on her list, preference wise), but if they surprise us and give her some huge wad of money, she would probably go there to have the name on her resume. She did get a callback audition, so fingers crossed.
Basically, you just have to play the game, cast a wide net, and try to network as much as you can. Try to get a good sense of your son’s level in relation to other musicians his age, and ask his teacher for recommendations of which schools might be a good fit. Try to apply for a reach school or two, a safety or two (people will tell you there are no safeties in music, but that’s not entirely true - you just have to seek out the schools that have good programs but aren’t necessarily the ones that are the best known), and then a few that are realistically on his level. And of course, that is different for every musician - one person’s safety is another person’s reach (Juilliard and Curtis are reach schools for literally everyone, though
).
We started doing college visits and trial lessons fall of my daughter’s junior year, so by the time she applied and sent in her prescreens, she already had teachers picked out that she had worked with and felt were a good fit. A few she’s had the opportunity to work with at summer programs and YoungArts, as well, so she already has a good working relationship with them, and we are hopeful that that will help when it comes time to decide on scholarship awards.