The effevescence you describe is most likely just CO2 in suspension. Yeast eat sugar & excrete CO2 & alcohol. Same process happens in bottle conditioning (carbonation) You will eventually lose the suspended CO2 if you age it; but you can always prime & bottle to make it sparkling again. Just prime like beer & bottle; you'll be good to go.
Backsweetening with a fermentable sugar will cause the yeast to start working again. They'll eat all that sugar & create more CO2, which makes more pressure in the bottle, too much sugar & the bottle could explode. If you want it sweet & sparkling, you'll need to backsweeten with a nonfermentable sweetener like splenda or stevia, then Prime & bottle.
The other way would be to bottle pasteurize. Here's a link to learn how to do that: