Che, I understand where you're from when you say you don't want to "annoy" your teachers for a rec letter because I felt the same emotion when I needed to ask my own for letters as well. But they are your teachers and they're there to help you and for someone to upload a letter for you takes less than a minute for them to do. Besides, you literally pay them to help you.
My math teacher I had for all 4 years of HS and I've used her letter for dozens of applications. She has one letter for me and uploads it or emails it whever I need, she just changes the content of it to reflect the name of what I'm applying for specifically. I'm sure you'll find that your teachers and counselors have a process that is just the same.