I mean I would not be so much embarassed about it. Presenting a defence in front of many professors is stresfull time for anybody and I personaly know some fluent people that were so stressed about it that they must took some antistress pills and of course I was making fun of them xD (I know those people personaly and they do not suffer with any anxiety or panic attacks, though they are more sensitive I would say). But in my case, nervosity usually helps my stutter (it only works for some long talks I have to give). Based on your writing it was mainly caused by nerves. I understand that it must felt like the most worst day of your life, especially in India where there is not much compassion with people who stutter, but you know, we are stutterers and this just happen to us, we are not in full control of our speech so this totally make sense and there is nothing to ashamed about. With regards to your future career, obviously you must realize that it may be hard as speaking is just not easy sometimes for us, but if you really want to pursue that career, go for it ! I myself will now be finising my PhD and lately, actually my stutter got so much worse. I think it was because of covid that I had. I have never imagined speaking could be so hard. So I am in a similar boat as you, now I have to decide what to do next and in those hard times I really did not felt like continuing. I sill havent decided, but I mean I really need to count with that speaking will just be very hard sometimes. I will go to speech therapy again in an attempt to gain some control over it and lessen the tension I get, it may be helfull but who knows. In your case if it is only because of the nerves, that still quite ok, you can try to lear how to manage it.