Hmm something wrong

I just finished bottling one grenache wine, and an apple concentrate with honey added.

They were both great at 1.000 or 1.002 sg, I bottled the apple stuff at this reading and it is now slightly carbbed and has an "off" taste to it. Not as good as it was before bottling.

As for the wine, it clearly lost its thickness or body it still taste good, but it doesnt taste as great as it did when its SG was 1.000. Its thin now! Is it very important to stabilize the wine ? Does fermenting completly dry take away from the body ?

This leads me to believe that they both oxidized and slightly spoiled. The grenache was kept under airlock at 1.002 - 1.004, a 3 gallon carboy, with 1gallon airspace. Since it was still fermenting, wouldnt the co2 blanket protect the wine from oxygen? It was only kept there for 3-4 additional days.

The "carbed" effect of the apple indicates either incomplete fermentation or residual gas. If your final gravity indicated complete fermentation then you certainly have residual gas and simply shaking the crap out of a bottle with a little headspace should clear it.

It sounds to me that the problem is that you prefer a slightly sweeter wine. Believe it or not, you can actually add a little sugar to the bottle and, once again, shake it up and you may be happier with the result. I know it sounds simplistic but it really does work.

The main symptom of oxidation is the "nail polish remover" smell. You don't give many details but sometimes the yeast can give up before the last bit of residual sugar is finished, and then nasty bugs take over and produce off flavours.(eg band aid) This will also tend to spoil the aroma and body.
I have no idea if this is your problem, just some possibilities.

A few people said that it might have been the lee's producing a sense of body, another contribution would be co2, and the residual sweetness.

I bet it wasn't finished fermenting, trapped co2 can affect taste as well as lees. spoiled dead yeast doesn't taste good at all. I always ferment to dry, I like it that way, then back sweeten for Daughter in law. Discovered people like blackberry better a little bit of sweet left so guess I back sweeten that too.Btw you can uncork all the bottles, dump into carboy, shack out the co2 , back sweeten and re-bottle for cost of corks.I have done that lets just say, more than once

Doesnt this shaking action oxidize the wine further ? Ive heard you need special equipment to degass wine properly like vaccums and wine whips.

I only degass kit wines and I use a drill mounted whip for about five minutes. I have a vacuum line bleeder but it's a pita and, imo, not worth the trouble. Bulk aging for a year under an airlock will do the same thing.

I'm with Summersoltice (well, metaphorically anyway!), I only degas kit wines because they are rushed to bottle. Letting a wine sit for about a year, and racking about 4 times (or whenever there are lees present after 60 days) seem to work just fine for degassing.

I think that if the wine was bottled at 1.002, though, it probably wasn't done. There was probably quite a bit of fermentable sugar left, and it just kept going in the bottle. You should never bottle a wine until it's been done fermenting for a while, it's completely clear, and it's stable.