I say it's yer intonation, mainly. Had a huge problem with it myself... If you're not comfortable doing it yourself you should take it to a decent shop and have them set it up for you. As soon as I got mine I took it to a ####ty shop and they didn't seem to know what to do with it. So I asked about it on here and got tons of help. Jaguars are fiddly and apparently never perfect. But they sound like no other...
I put a Gotoh Nashville Tune O Matic bridge on mine and adjusted the neck and it's as good as gold now. It's all quite simple to do, I guess. Just make sure you've got the patience to keep taking the neck off. I tried to adjust the truss rod with the neck still attached and it just cracked the lacquer under the scratchplate. Invisible but foolish, like.
There are loads of little tips on this site and many others on what you can do to make your Jag easier to use, it's worth looking around.
The difference between MIJ and CIJ is, allegedly, nothing but age. MIJs are older. Hence cooler!
And I doubt changing the strings will do anything. Think I got the bridge for about £20.
Welcome to hell