D.,
First off I want to say Wow. I am just like you in many different ways. Is there an easy answer of how to juggle everything at once, probably not. The one thing that I can suggest is to try and sent a timeline. Yes, the baby needs you when they need you but where there is a will there is a way.
Who watches the baby when you are in school? Is there any way they can watch the baby for a little bit longer so you can either get homework done or run errands or just have down tiem?
As they say, when the baby is asleep is the best time to do evreything you need to do.
I work a full-time job, have two kids (ages 6 & 3), and do online schooling. The way I do things is set times. I know that I will be home from work by 5:30 so dinner is priority #1. Then, I spend some time with my kids until I put them down at 8pm. After 8 I do a few errands (for example, Monday nights I pay bills, Tuesday, I dust, Wednesday, maybe homework pending to be done, Thursday - sorry, but it is Grey's Anatomy, Friday, finish up any little thing that needed to be done. Organization is definitely key. I live out of my planner because that has dates and schedules and appointments that need to be kept.
I wish I could say it gets easier, I think it gets manageable when you get the hang of things. Most importantly, just enjoy the baby now and appreciate it. Understand that the dishes will always be there tomorrow!