What a great question.
First of all, I want to clarify that it is not the emotions that are causing your pain. Rather, it is your body's response to the emotions in the form of tension and anxiety. You mentioned that people upset you which leads to pain and this is something which you fear. The fear is the primary reinforcer of the pain so interrupting this pattern of reinforcement should be your primary goal.
Yout primitive brain is interpreting danger when you experience certain emotions like anger and/or sadness. When you experience this, first create safety for yourself by breathing and sending a message to your brain that you are safe and not in danger.
Once you have this bit down, then you can absolutely practice feeling your emotions, ideally in your body. Regulate your anxiety and then focus on how anger/sadness feels in your body. What's the sensation? Can you just focus on how it feels and allow yourself to feel it?
If you need to remove yourself from a situation to practice one or both of these exercises, do so by all means. After all, what is more important than feeling safe and caring for yourself?
This takes a lot of practice so don't get discouraged when it doesn't work. Just remind yourself that you're learning how to care for yourself in a new way and that this takes time and patience.
Best of luck!
Any advice or information provided here does not and is not intended to be and should not be taken to constitute specific professional or psychological advice given to any group or individual. This general advice is provided with the guidance that any person who believes that they may be suffering from any medical, psychological, or mindbody condition should seek professional advice from a qualified, registered/licensed physician and/or psychotherapist who has the opportunity to meet with the patient, take a history, possibly examine the patient, review medical and/or mental health records, and provide specific advice and/or treatment based on their experience diagnosing and treating that condition or range of conditions. No general advice provided here should be taken to replace or in any way contradict advice provided by a qualified, registered/licensed physician and/or psychotherapist who has the opportunity to meet with the patient, take a history, possibly examine the patient, review medical and/or mental health records, and provide specific advice and/or treatment based on their experience diagnosing and treating that condition or range of conditions.
The general advice and information provided in this format is for informational purposes only and cannot serve as a way to screen for, identify, or diagnose depression, anxiety, or other psychological conditions. If you feel you may be suffering from any of these conditions please contact a licensed mental health practitioner for an in-person consultation.
Questions may be edited for brevity and/or readability.