@keepmeonpins: You're certainly not doing anything wrong!

For myself, I only blow out my attys if they start to under-perform (ie. vapor production seems to go down, taste starts to suffer a lot, gets too hard to draw on). The only time I would suggest you *have* to blow out the atty between flavours is if you're going to use a flavour that will taste terrible mixed with the last flavour. More often than not, the old flavour will completely disappear in about 4-5 drags. Until then, you'll have an interesting combination of the old and new flavours, which is sometimes a good thing.
Also, tobacco flavours tend to be VERY resilient in terms of what other flavours can be mixed well with them.
As for how you're drawing, the only right way is the way that gives you the most satisfaction. Try drawing a little longer, a little shorter, little harder, little softer. Have fun experimenting with it to find the method that works best for you. Most people suggest a light, 5-6 second draw. Though whether you get the best results drawing into the mouth first or straight down your throat (direct-inhale, as some call it), is hotly contested depending on what you want most (between flavour, throat hit, and vapour). Short answer: play with it, and do what works best for you... THAT will be the right way!
EDIT: Also, as long as the filler in the cart is wet, and making contact with the atty bridge, the atty can't get dry. The extra drop or two certainly won't hurt (and will definitely keep the flavour AWESOME), but not strictly necessary as long as the cart filler is still wet where it contacts the bridge.