Hi Omair312 and thanks for your post.
I have a few initial comments in response to your observations that might be useful. 1. Rest did not aid recovery at all & 2. You've noticed a 'lump' in the left upper trapezius area and perhaps a slight 'lean/elevation' of the shoulder. This could indicate a block to the resting position of the left scapula (shoulder blade). If the scapula is not held/stabilised in the correct position during an activity then the activity of the muscles that rely on that stability will be reduced. It sounds like in your case the scapular might be slightly elevated on the rib cage. This would give more prominence to the trapezius on that side and perhaps would involve some overactivity of the muscles attaching to the 1st rib also on that side.
What might the scapula not be in it's ideal position? It is possible the lower fibres of trapezius are not firing at the right time and therefore not keeping your scapula depressed at the right time or your Serratus Anterior muscle is not controlling the scapular during presses and overhead movements (can be due to long thoracic nerve issues, LTN, from the neck C5-C7). In the case of a LTN issue you would also notice what is called a 'winged scapula' in a press-up position.
The shoulder itself is unlike many other joints in that it is not really connected via a joint (like this hip) to the skeleton (it only connects at the sternum via the sternoclavicular joint). It basically floats on the ribcage and requires a complex balancing of the muscles that stabilise/control its position in order to allow the deltoids to do their thing. If the scapula is not stabilised the deltoids will not function at their full capacity (nor will anything else relying on it).
My advice would be to have an physical assessment with someone who can observe the movement of your scapulae as you move your arms from by your side to various overhead positions, paying particular attention to any asymmetries (difference between the right and left sides). Ideally that person would also be able to assess your neck mobility to see if something needs releasing there and your general thoracic spinal mobility to see if it can extend enough to allow the scapula to move effectively throughout movement. For now I would back off the overhead activities and any other exercises that require a lot of scapula stabilisation e.g. dips, triceps push downs, chins, even bench press until we get more information. From there we can see what you need to work on to get the base stable before you then reintroduce more traditional delt type workouts. So it's a couple of steps back to sort out the base, and then carry on upwards from there.
Feel free to ask more question and post any further insights along the way
Many thanks
PB