Hi Xei,
I know exactly what are you talking about.
I graduated from Cambridge with a 3rd class degree in NatSci in 2009. I was good at problem solving and had good 1st and 2nd year results, but all the third year exams are concentrated on essay writing, which just killed me right away. In July, I was told that I cannot continue to the 4th year of tripos and have to graduate in the next month. I always wanted a career in academic research but all I got at that time was something like "you're not cut out for it" or "you've got no chance".
Still didn't want to give up, I started looking for job positions in academic research. I can't remember how many of my applications had been turned down but finally I got a job as a research assistant working on software development for some biochemistry stuff. I worked for 2 years and had a few second-author publications. Then I used my publications along with my poor undergrad results to apply for masters. The odds were still not good and I was rejected many many of times. In the end, I was lucky enough to be admitted by the University of Manchester in master of computational biology. From then, things started to get better. I got good marks in the masters course and now I am in my second year of PhD at Uni of Manchester.
So Xei, as long as you're sure of your competence in this area, there will be a way out. Everyone said a 3rd is impossible for academia, but I just found out myself it is possible. The fact is that exam results don't always reflect your ability. Many universities and institutes are aware of this.
I'm not sure if this can be of help. I can tell you more.
Best