@Mr_NutterButter assuming you dont have outstanding abnormal metabolism related errors, then your feeling of throwing up may be either poor breathing (cramped, partial, holding your breath or combination) or overly cramped (lower) core (stomach area) muscles while doing exercises. Or both.

Breathing you can control to get good, consciously and over time subconsciously.

The stomach muscles being too cramped (not muscle cramp, but just using them too strongly) from trying to hold proper posture, fixable with either proper posture so that you dont need to use them, and train them as well properly and slowly. Stomach muscle issues SHOULD resolve automatically over time as well, but conscious training may be required should you not desire to feel like vomiting. Cardio/core training combinations should help with that, if you reach a desired outcome (such as "please dont feel like vomiting after training"), then you can stop the cardio/core p much.

and the usual: be well hydrated before and after training, dont have a fresh recent LARGE portion of food in your stomach before starting exercise, deep breaths between each exercise (and pauses between exercises).

If your gut becomes oxygen deprived (either lack of blood flow from muscles being too tense, or poor breathing and circulation), it is always guaranteed to feel nauseous. Everything i mentioned on its own can cause poor oxygenation of the gut area and be enough to give nausea. If there are multiple problems simultaneously present, it becomes worse nausea. Each issue mentioned has a in-depth or simple solution.