I used to take synthroid but I switched to levoxyl for monetary reasons. I have not suffered from this with either drug.Though I don't drink soda much, diet or otherwise. My mother takes synthroid and has no issue either.
Your stomach problems may be unrelated and completely coincidental to taking this medication. I did a little research and came up with a few possibilities.
If the soda you're drinking is non diet, you could be in the early stages of diabetes and should get tested for that. Since synthroid regulates the gland that regulates your metabolism, and you're now showing problems metabolizing sugar, you may need to get your insulin levels checked.
Another possibility is that you've developed a caffeine allergy. Most sodas are caffeinated so try drinking a non caffeinated, carbonated beverage and see if you still suffer. If you're fine, it's not the carbonation, but the caffeine that's your nemesis.
If your insulin levels are fine, and you get a test to see if your allergic to caffeine and it comes back negative, then you may just have to give up sodas which wouldn't be an alltogether terrible thing since they're pretty much terrible for you. Regular and diet.