Effervescent cider?

I have 3 one gallon jars of cider that I started July 31st. 8 day primary at which point I thought they tasted like vinegar. I racked to secondary and tasted again 2 weeks later. Vinegar taste had been replaced with more of a yeast flavor, my wife described it as "beerish". Today, I took a small sample and each is a slightly fruity dry wine with a very fine effervescence. Not sure how I accidentally accomplished this, but I sorta like the mouth feel of it.

My plan was to backsweeten, bottle and toss into a cold fridge. Would I lose this effervescence if I did this? How can I keep and/or recreate this carbonation level? If I continue to let this age under airlock, will I eventually lose this carbonation level?

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:
https://www.homebrewtalk.com/f32/easy-stove-top-pasteurizing-pics-193295/
Regards, GF.

Guess I wasn't really clear in my questions...I've carbed my cider before, but never with such tiny bubbles. When I've bottle carbed, the bubbles have been small but not nearly as tiny as these. It has been under airlock the entire time so not sure how the CO2 built up like this. What determines bubble size and how would I recreate this effect?

The carbon dioxide built up by being constantly exposed to it. The yeast have been bubbling it through the liquid for the past few months and some of it dissolves. When you disturb the cider you knock it out of solution.
The reason the bubbles are small is there isn't very much carbon dioxide in solution.