My name is Matthew from the website organicwinefind.com. I also live on an organic vineyard in France so I get to see both the farming and winemaking process and just what a difference it makes to be organic! I'm going to try and answer your question without writing an essay, but there's lots to consider when buying organic wine, so I apologize in advance if this is a bit long-winded!
As you probably know there are two stages to making wine :
1) growing the grapes and
2) making the grapes into the wine
Organic grapes are grown without the use of synthetic fertilizers, pesticides and herbicide, so when you see "made with organic grapes" on a wine bottle you know that you're getting wine that doesn't have any nasty chemical residues from the farming process. Farming organically is also supportive of the health of the farmers (who don't have the danger of working with toxic chemicals), the people that live around (its pretty scary seeing conventional farmers spraying their grapes in hazmat suits and oxygen masks!) and the environment.
Regarding the environment, years of farming with chemicals destroys the life of the soil leaving it barren, grey and erroding. The ground water gets polluted and runs into local streams and rivers killing fish and other species. Its a vicious circle because dead soil can't support growth so requires more chemicals etc etc. Organic farming (and especially biodynamic) by contrast, nurtures the soil and increases biodiversity. Grapes that are grown organically are generally more flavorful and alive with natural vitality. In the hands of a good winemaker they can make incredible wines. (Fun fact: Of the 50 most expensive wines in the world 31 of them are made with organic grapes.)
So thats the grape growing part and IMO the most important aspect of organic wine. Now for the winemaking.
The key decision you need to make when you buy organic wine is where you stand regarding sulfites. Sulfites occur naturally in the winemaking process and they are also manufactured and added to wine as a preservative. This has been done since the 15th century. Sulfites keep the wine fresh, they balance the wine's taste and support it to maintain its flavor intensity. But sulfites can cause minor allergic reactions in some people. Even so, most organic producers still chose to add sulfites because it is widely considered to produce the best wine. Organic wine that has sulfites added is labeled "made with organic grapes". Legally, it cannot be labeled "Organic Wine" because the added sulfites are man-made.
Sulfur free wine has to be made with extreme care and by a very skilled wine maker. Only a handful of winemakers in the world are really making outstanding sulfur free wines. Without preservatives it also has to be carefully stored and at the right temperature. If its not well made or properly stored, it can end up tasting weak and funky. At worst the wine can go off and taste downright awful!! Its likely that the wine you tasted was sulfur-free and may not have been in good shape. There are some good sulfur free wine makers but you have to chose carefully, and you have to know in advance that its going to taste a bit eccentric!
Wine labelled 'Organic Wine' in the US is sulfur free, and as you experienced yourself, some of it isn't doing organic's reputation any favors!
Personally I prefer to drink organic wine in the "made with organic grapes" category because 1) your get all the benefits of the amazing organic grapes without any of the downside of funkiness 2) you have a much wider selection of incredible organic wines from around the world. 3) IMO the most important part of the 'organic' equation is the farming because of its health and environmental benefits. I find that the small amount of added sulfites (and most organic producers add much much less sulfur than conventional wine) is a very small price to pay for great organic wine.
The more people say "this organic wine is great" the more organic farming will become the norm. The more people find it unpleasant, as you did, the further away that reality becomes. So I say "bring on the sulfur" if it helps people enjoy organic wine!!
In terms of some recommendations, there are so many great organic wines red, white, rose, sparkling to choose from that you may want to come on over and use our search tool which allows you to put in you preferences and get a list of organic vineyards that match. You may also like to watch some of the reviews at www.organicwinereview.com and www.organicwinejournal.com to get some ideas.
Hope this is helpful!
All the best