I'm always confused about orders, because lately the number of orders is really growing and I don't have enough time to keep track of everything promptly anymore. It seems I am required to find and hire a new employee to oversee this process.

Well, the bigger the business, the more processes, tasks and more employees are required to fulfill them.

If the load is critical, maybe a new employee will solve the problem. But still I recommend you including learning more about Magento extension for order management, creating notes and other labels on amasty site. Of course, everything is individual, but I found it very useful to control the chaos that used to happen on the orders page. It's especially handy now to prioritize tasks.