I do an hour and a half of yoga everyday, and have done so fairly consistently since age 18. Prior to smoking, I used to be an extreme exerciser, I ran cross country and biked, on average, about 200 miles per week, often more.
Vaping doesn't make me sweat *more* but I believe it is one symptom of quitting tobacco, so that might be your issue (though if you weren't excercising at all, kudos to you for starting, it will make your quit easier IMO as you will slowly and steadily a) feel better-- endorphins! and b) wash those toxins out of your system.) But my point is, if you've JUST started to exercise, I'm not sure the shortness of breath and/or sweating is connected to anything other than *starting* to exercise, period. Of course you are going to be short of breath. I am finding my speed/endurance other factors with yoga is greatly increased even after a few weeks not smoking, and am looking forward to mountain climbing season.....
However, if I buy a juice that makes my lungs feel heavy and/or I find gross, I tend to not use it anymore, as the main reason for vaping, for me, is to *improve* my lung health, not worsen it....
If you are concerned, you could try a different type of inhale for a bit, *only if you want to* in through your mouth, out through your nose, no lung activity. It sounds wierd, it feels weird, but you may have been a heavy enough smoker that you want to do that for awhile.... You could also try different flavorings, different PG/VG ratios, and consider wattage and other issues. I'm currently vaping some vanilla flavor, which makes me pretty hoarse, but I ran out of juice and I'm not driving at the moment due to a health... thing, and the meds I'm taking. I don't inhale that stuff as much as I might otherwise, and I don't much care for it, but I also don't want to drive to my local vape shop, and hit someone, either.
Anna