I have a few thoughts on this based on my experience (which may differ to yours and many others' of course!) and I'm typing quickly before going out to lunch so I hope I make sense but I resonated with your post so...
I'm glad you're working out what feels best for your stomach! An upset stomach and strange heart feels are horribly disruptive to life. I experienced both those for decades (I'm 31 and I've only felt better in the last few years).
I have a mother and Grandmother with Coeliac disease so they tested me and found I had the antibodies etc. (this is under fire in the last week or so though! I won't go into that whole story here/now...) so I went gluten free but continued to be veg and in fact vegan for most of that time (barring the last 3 months of experiment with fish). It's definitely doable but does take a sometimes frustrating amount of planning.
Buckwheat bread is fantastic (and you can make GF bread sourdough like 'normal' bread), chickpea flour is nutritious and as proteiny as wheat, and...oh. Lots of things! A lot of readymade GF products happen to be veg/vegan because they're catering to allergies in general, too.
That said, it's also possible the stomach pain is coincidence (could you be reacting to something else in the bread? Something else entirely?) and/or you don't react well to wholegrains (I have a few friends in that camp) in which case you could try sourdough ones or soaked ones or just no wholegrains at all, but don't need to avoid gluten.
You might end up avoiding all wholegrain AND non-wholegrain wheat but that's still easier done than gluten, which is pretty extreme and has trace elements in all kinds of useful foods
. So it's good to really investigate fully what's going on so you don't limit yourself more than you need to
Weakness could be a deficiency (from gluten intolerance etc. or just general diet) but can also be anxiety. I experienced panic attacks for many years (never do now. Ever!) and it felt a lot like you were describing. Which could still be dietary but could be something else.
Do let us know what your doctor says after you bring it up with him ?