The typical signs of illness to watch out for are basically the same and this is regardless of how new the budgie is. Even without not yet having a good understanding of the new budgie's personality/behaviour, it's still possible to detect the signs or symptoms exhibited when something is not well with the bird, this is what quarantine is for, to observe, try to make a connection and take the first steps in bonding to later on begin training.
From your description, your new budgie seems to be perfectly fine.
Given how busy you currently are with your studies, you can continue to spend quality time near both of your budgies' cages, talking to them and occasionally offering them a little treat to eat through the cage's bars.
This will not be taking a whole lot of time away from your books and you can take advantage of some of your little breaks to tend to your budgies.
Try not to feel pressured in reaching a breakthrough in trust/taming with your budgies within your reading week.
Once you are less busy, you can devote more of your time in working on bonding with your budgies and keeping the training sessions in a consistent manner for better results in the long run.
For the introductions, keeping both cages close by and allowing your budgies the required time for them to get used to one another would be good.
You can keep this arrangement for the first couple of weeks and depending on how they respond to it (if they both seem eager to truly meet and have a closer interaction) you can then allow them to have supervised out of cage time together. I realize this may not be easy especially if none of your budgie are tamed or used to having a routine where they will have out of cage time in a safe bird proofed room.
You can check this link for detailed information: https://talkbudgies.com/new-budgie-ar...o-budgies.html