I just solved the problem. I would delete this question but the reason for the "Read-Only" state has me confused. Anyway, this is what caused the problem and how I solved it.
After adding a new field, I wanted to sort the data for easier data entry. However, after I sorted the data on a field that has a look-up drop-down, the table became read-only.
After I removed the sort, I could change the table again. I have no clue why Access behaves this way. If anyone can provide any insight into this behavior, I'll update my answer. If you see this issue, just remove your sorts and see if it fixes the problem.