Hi!
Although at the speed you're at now, it doesn't make a difference, you shouldn't be dropping your left arm into your lap when it's not playing. Use that time to get it in position for the next chord.
Okay, here's what my teacher says in general: Fingers provide support for the power, they do not give the power. The power for pressing the finger down should come from the knuckle that joins the finger to the hand. Trust me, it feels much lighter. When playing the thumb, make sure your using the tip and not the side. On the scales, lead with your elbow. Imagine you are a violinist and that you are bowing across the keys. Your fingers should just follow as your elbow pushes or pulls them across the keyboard. Make sure you're not reaching your thumb under but simply letting it fall into place beside the other fingers. The elbow motion should be smooth, there shouldn't be little jerks that come from flipping the thumb under.
A nice exercise for relaxation: Play with one hand alone, and with the other hand, support the palm of the playing hand with the index finger. You should be able to rest the whole weight of the arm on the index finger and still play. If you were to take your hand off the keyboard, and then remove your index finger, your arm should drop to your side. Don't push it down, just let it relax.
Lastly, and the thing I need to work on the most myself: Listen! If the scale doesn't sound even, slow it down and figure out why it doesn't sound even. For example, your third and fourth fingers may "clump" together. The first step to fixing the problem is diagnosing it. You should be able to cure problems with unevenness by playing from the knuckles and guiding with the larger members of the playing mechanism (i.e. the lower and upper arms).
Don't expect it to happen overnight. You must pick a problem spot in the music and work it every day until you reach the point of diminishing returns. It's amazing how much progress daily attention can get you. I know this is hard, but trust me: forget about your exams. This is not about you passing a test. This is about you becoming a better pianist. If you trust in yourself and practice consistently, your teacher(s) will see progress, and they will understand that you are willing to work towards bettering your skills. Teachers see so much we as students do not and are amazingly accurate at assessing a student's skill level and progress (assuming you're doing this for some sort of reputable music school). Select the Bach and the Chopin now (I suggest the Chopin Waltz, and out of my personal preference, the Bouree). Find recordings of all of them, listen, even dance along. Get the music inside you and learn it because you've come to love it. Show, through your playing, that you love what you're doing and that you truly believe the music is worth your time. That's what matters. Best of luck.