The problems you describe are very common. PrepLounge has great case prep materials, and you should also look up Victor Cheng's 6 hours of free videos on YouTube if you haven't yet. Besides this, the best and fastest way to improve quickly is to work with a coach, who will be able to give you personalized and actionable coaching points. You still have 5 weeks, so that's plenty of time to get better - but don't discount how much work is involved to really become good at case interviews.
As for the timed exercices you identify below, the key will be time management. Good luck