As someone who has experienced considerable generalised anxiety (GAD) throughout the course of my PhD (finally completed in 2015!) I can thoroughly recommend seeing a therapist to discuss how you are feeling. In particular, I have found CBT therapy to be very helpful in both understanding better the biological reasons for my responses to things (fight or flight as others above have described) as well as techniques to better handle it.

However, I would also say that there is no shame in seeing a psychiatrist, which I also do, to keep an eye on the medical side of things. While a GP can be helpful initially, nothing beats seeing an expert who has seen it all over the years and can reassure you (nothing you say will shock them!). In particular, for me, a combination of the aforementioned CBT therapy and medication has let me lead a relatively normal life and I now have a wife and young family to add into the mix!

It truly can (and will!) get better and the first step is seeking some help. It is important to realise that academia can be quite a toxic environment (at least I have found) and that you're not necessarily 'losing the plot'. What you're feeling, especially with your supervisor leaving etc, is in all likelihood a normal response to the situation... and the responses of some of the people around you, such as telling you to 'pull it together' is actually abnormal! Put all this together and it is a recipe for feeling quite rubbish :-)