I got my courses figured out. I got some valuable information from the Bio Department, and I hope this helps anyone else having a problem:
"Available seats are reserved and you do not meet the reserve capacity criteria" - a number of seats in each course and each section are reserved for students in specific programs, so the section may appear to have space, but the seats are not available to everyone. For example, Bio 1A03 will have a certain number of seats reserved for students in Biochemical Engineering, Science Level One, and the Arts and Science program. If you are a Science Level One student but the only seats left in the section you are attempting to enroll in are reserved for Arts and Science students, you will need to choose a different lecture section in order to register. Typically Bio 1A03 fills up more quickly in first semester and Bio 1M03 fills up more quickly in second semester, so try switching them if you are having difficulty.
I had to switch up which bio class I took in Fall/Winter (which was a bit of a hassle to do - had to rearrange half my classes), but it finally worked.
Why don't they just post this in Mosaic? It would save all the departments a ton of time. I had to contact two separate people just to get this info. So it's not a tech error on Mosaic's end, just a lack of communication regarding the capacity error message.
Thanks for the replies! Glad I got everything sorted out now.