@HemanBrew, you can try and put a bottle in the refrigerator for 24 hours to see if you notice any increase is carbonation. While it is true more CO2 is absorbed/dissolved into the beer at a lower temperature, I'm not sure if one could actually detect a change with such a small amount of headspace. Though, as mentioned before, you have too much headspace.
Have you ever dropped a warm bottle of beer or punctured a warm can of beer? If so, what occurred? I suspect you saw a lot of foam as the CO2 was releasing out of the beer. My point, warm beer does have CO2 dissolved into it.
Reasons why bottles do not carb could be : 1) caps do not seal, 2) the yeast has flocculated out so there is no yeast to consume the priming sugar, 3) there wasn't enough priming sugar used, 4) the bottles were conditioned at a lower temperature that caused the yeast to go dormant, and 5) the bottles were condition at a high temperature that effectively killed any remaining yeast.
Any one of the above, or combination there of, could cause the bottles not carbing. Heck, maybe you miscalculated in the amount of priming sugar you added. Though, with you commenting how sweet the beer is tasting, I suspect that is not the case unless you had already a large amount of unfermented sugar to begin with.
I would review your process by listing the steps you took to see if you can identify what occurred. I'm curious, in those bottles that you opened (or at least the recent one), did you notice any sediment at the bottom of the bottle?