Kefir is Time Tested food - I am from exUSSR and we always used Kombucha & Kefir with great health benefits.
Just give it a time - you are in transitional period and sometimes it gets worse before it get better!
Also, I am wondering, what dosage you take?
If you started on low amount - you have to increase it everyday - until you take a Liter (a 1/4 of a Gallon) a day - to receive medicinal benefits.
Small amount could push Candida out to other parts of the Body - hence, flare ups in other organs...
As for Dairy Kefir - I have some concerns too... since Commercial Milk is not the same as organic Raw Milk fresh from farm - on which I grew up.
But, I have solution to the Milk Problem: I, mostly, make my Kefir from organic Soy Milk!
You, also, can make Kefir from any other substitute Milk: rice, almonds, pumpkin seeds etc...
So, you can stick with your Veg Diet and still receive your Pro-biotic Bacterium - daily.
Periodically, I use Real Milk or Mixture of 50%/50% of real milk and Soy milk.
Reason: Kefir Grains are Milk grains - not Soy grains... so, they will stop growing on soy and eventually could die-off...
My advise - keep taking it! It's the best thing for your body!
All my Family and all our Friends drink Kefir & Kombucha!
We cured many health problems by drinking kefir!
Once you cleanse-up & re-populate your intestines with good bacteria - you will feel much better!