I don't think Cameron is in favour of leaving the Convention at all (since his speeches are constantly peppered with references to the rule of law), its certain factions within his party.
And I don't see how its possible without leaving the EU...it might be that some are confusing general EU law with human rights, when some of the headline grabbing immigration cases (which seem to nearly always be decided in favour of the immigrant) actually involve the freedom of movement right being implemented extremely purposively when combined with a fundamental right, such as the right of non-discrimination.
Anyway, there are two separate sources of human rights at EU level - the basic level Convention on Human Rights, and the EU Charter on Fundamental Rights, which applies to all EU Member States when they are applying EU law, and is more extensive and modern (e.g. it covers rules on data protection) than the ECHR. It is, for example, where your Freedom of Information rights come from.
Its not true to say that the UK is the only country that struggles with the issue of human rights from a political perspective; Hungary is constantly being threatened with legal action for introducing changes to its Constitution which the EU doesn't like.
Incidentally, both Hungary and the UK traditionally do not have an official single Constitutional document (although Hungary now has a constitution). I find that in itself shocking, and think this an issue which should be made more of. On the one hand, you can argue that the UK is gaining from the flexibility provided by not having one, as it is consistently the most economically successful EU Member States, on the other hand, even the US has one - although in other areas, its approach is far removed from that which most European countries adopt.
In practice though, the UK is one of the best EU member States at implementing EU legislation correctly. Although that's not to say there aren't areas of concern, there are certainly problems in other EU Member States as well - the ECJ is full of such cases.