People have pretty much covered it Jazz, you're right that priming=sediment, as discussed this is no bad thing. Commercial beers that say 'bottle conditioned' will have some sediment, the rest have been filtered and gassed up from a CO2 bottle. It is possible to do this to homebrew if you're using cornelius kegs but otherwise not, many think the bottle conditioned is superior anyway.
If you have a look at some bottle conditioned beers next time you're in the supermarket you will notice that the layer is only very thin, probably a lot thinner than you're getting in your bottles. You can reduce your layer to a minimum by:
1. Giving plenty of time to clear in the FV, or transferring into a second FV and allowing settling there
2. You can encourage this settling by using finings if you want, a period in the cold helps too, at this time of year a few days outside or in an unheated shed/garage will stop yeast activity and encourage clearing
3. Use the right yeast. The term your looking for is 'high flocculation', as mentioned Nottingham would be a good choice to replace your kit yeast. It should give better flavour than most kit yeasts but also tends to drop well and form a nice solid layer because it is a high flocculation yeast. This should mean it clears better in the FV so less yeast goes into the bottle and also the layer in the bottle won't disturb as easily so you'll be able to pour more. Another popular dried yeast is US-05, which is good for keeping hop flavours in your beer but a bit rubbish at settling and forming a good layer - I know I'd like to find a yeast that keeps the flavours like US-05 but settles like Nottingham.
4. Give it time in the bottle. Not only will your beer improve with time but the sediment will get a bit more compact and less easily disturbed. I recently drank a bottle of a year old ESB that had been made with yeast grown from a bottle of Fullers Bengal Lancer. The combination of a high flocculation yeast and a long time in the bottle meant that I was able to pour the whole bottle and the yeast layer just stayed stuck to the bottom.
At the end of the day a bit of yeast in your beer will do you no harm, people buy brewers yeast from health food shops and use it as a dietary supplement, apparently "it offers 9 vitamins, 14 minerals, and 16 tissue-building amino acids ". When trying to persuade sceptics that homebrew is good beer though it is handy to be able to serve up a clear one.
: