Good advice and I agree that obstructive sleep apnea (you don't have to be asleep) may be you problem, and it may be due to the weight gain. I am a bit older than you and never even a heavy drinker... somewhat into fitness in my middle age years.
I noticed that I was waking up felling like I was suffocating. I would also notice it when nodding off watching TV or the like. At that point I weighted as much as 250 pounds on a 6'5" frame so overweight, yes, but except for my waist line and it turns out my throat,, I wasn't REAL FAT.
I purchased a simple Oximeter clip (clips on the finger tip, cloths-pin like) fro about $25. I give a quick readout of Oxygen Saturation (its named purpose) and also the heart rate. I consistently found my O2 level in the 95% or better range when awake. So, I put the meter on my night stand and when I woke up feeling the problem I clipped the meter on and found readings in the upper 80% range, not good. I discussed with my Primary Care and was tested at home again with a recording O2 meter, very inexpensive so not a stopper even if paid for out-of-pocket. That confirmed a problem. It could have ended there, but I have Medicare and Secondary Insurance so I underwent the expensive Sleep Study at the hospital. That resulted in my being diagnosed with OSA. I discussed with the attending Pulmonary Sleep Specialist and he agreed losing some weight might fix the problem, so we put off putting me on a CPAP mask, another big cost and not great solution. I lost 25 pounds and no more trouble with that - by the way I also suffer from Atrial Fibrillation but that was unrelated to my breathing problem.
Congratulations on your sobriety. As to the question you asked at the end of your post, I would say that you probably need both things, rather than one or the other. Everyone needs a healthy diet, an appropriate exercise plan, and not to smoke. There is no one on earth for whom any amount of smoking is physically healthy. Because of your current symptoms, however, I agree with Red_Star that you should see a medical doctor. If I were you, I would postpone the exercise plan until being medically cleared. In the meantime, though, you can start making your plan for smoking cessation (decide on a quit date, what support you will have, etc.), and you can make some of the more obvious adjustments that you need to make to your diet. Good luck.
Wake up gasping for air is a symptom of sleep apnea (sleep disorder in which breathing repeatedly stops and starts) as well as heart failure (the heart muscle cannot pump blood around the body effectively/properly). See your doctor ASAP.