I have studied singing for over 10 years and have taught
ESL and I have never had someone ask me about this. It is an interesting question. When you speak a different language, especially if it is quite different from your native language as English is quite different from Vietnamese, the new language should definitely feel different when you speak it. The tongue, jaw, and lip positions for sounds that are not in your native language might feel strange at first when you do them correctly. For this it is good to practice the sounds repeatedly with a mirror so you can watch your own mouth to make sure you are doing these new foreign sounds, and not something more like your own language (which will feel more familiar/comfortable).
As far as breathing, I agree with Angelika that it is not something you should think about too much. When you speak Vietnamese, do you take big breaths before you speak in normal conversation? I would guess not. The same is for English. I would think that if you are taking big breaths before you speak a sentence, that irregular breathing is probably why you are feeling uncomfortable and out of breath at the end of a phrase. I have studied many foreign languages for singing and I really feel that English should NOT feel different in your breathing, only in your mouth/tongue/jaw. Everything I have learned about breathing while studying singing is NOT something I would ever apply to speaking in normal conversation, only something like theater.
I have made a website related to this topic, with video How-Tos on all the Sounds of American English. You might find it helpful to work through some of the sounds:
RachelsEnglish.com.
It's hard learning a new language! I am sure your hard work and attention will help you a lot along the way!
Rachel