I'm not as familiar with the details ae are you. However, this sounds like normal stuff in politics (and life). Politicians (and people) often say what is convenient at the moment in rationalizing their actions, as opposed to the whole truth. It does appear that Cameron has lost some of his political support in the UK, as you suggest. At the same time it also appears the there are underlying issues in the UK towards EU governance, and that similar issues lurk perhaps beneath the surface in other EU countries as well.
I think the basic dilemma attendant to the historically remarkable success of the EU over the past generation, a success that was often facilitated by bypassing some key underlying political issues, remains the ultimate resolutions of those very issues regarding sovereignty. A few decades ago the external political and economic threats tended to make delaying the final resolution of those matters look very justifiable. Now, in the absence of such immediate threats, it appears that resolving them will be very difficult indeed.