Whether people know it or not, everyone has some sort of intolerance to gluten, whether mild or severe, some don't notice symptoms, some notice lots of symptoms. I actually don't think humans should be eating it regardless - I always think back to tribes etc.. would they have made bread? Would they have eaten sugar? (Apart from fruit)...Not so much. It's very much vegetables, fruit, berries, meat and fish etc. so it's probably no wonder why after all those years of our ancestors eating "real food" that our stomaches just didn't evolve enough with the evolution of food containing gluten or sugar or whatever it is that's in them these days!
I'm currently being tested for Coeliac's, and what you've described to me is similar to what I've had in the past. Dizziness, or vertigo has popped up at random times, from waking up, to going to bed that night, or even longer. Weakness though, I usually experience if I haven't eaten yet for the day. However, I am also addicted (like most) to sugar. And all the studies I've done into that shows that sugar, and even the period of when you're trying to give up sugar, can elicit all sorts of physical and mental symptoms: from mood swings (although gluten does that too so it's a double whammy) to disturbed sleep, lack of absorbing nutrients, highs and lows, lethargy, shakiness and dizziness etc.
I advise doing your own research into both though, because it's always helpful to fill yourself with enough knowledge to take to the doctors.
The fact you've had it for three months is disconcerting though. Have you had an MRI? What other tests have you had done?
My only advice at this point would be - don't do an elimination diet just yet - if you want to put coeliacs out of the way, go and get an endoscopy done (don't bother with the blood test, it rarely comes back accurate) and if you have coeliacs, you'll be advised to go on a gluten free diet, and hopefully be directed to someone who can offer yearly support.
If you don't, then I would suggest an elimination diet.
Someone had suggested I eliminate gluten, then a few months bring it back into my diet to see if the symptoms reduced/came back - I've never had more torture in my life, but it proved my suspicions right. So eliminate gluten (if you're not coeliac and still can't figure it out) and eliminate sugar or other additives. Try a non-processed food diet and see how you "feel". There's so much crap pumped into our food these days, and unfortunately a lot of us just aren't as aware of what we're eating, as we would like to be.
Don't stop looking for answers though, it does not sound good at all! Make sure to seek second or third opinions - doctors aren't always right unfortunately
I wish you all the best.