In addition to all of the above:
Make the cast more out to the side, i.e. away from the body and your fly rod, then you would normally cast with dry flies.
When that lead-eyed streamer is coming forward, you do not want it hitting your rod (possibly cracking it) and you don't want it hitting your head or back.
So instead of casting with the rod near vertical, cast with the rod tip a little more out to the side than normal.
Make a slow backcast, and wait until the weight of the fly fully extends the fly line, and you will feel that weight actually start to bend the rod, then make the forward cast. Yes, slinging the weight forward, as someone already described.
In normal fly casting, you are throwing the weight of the fly line, and the fly just goes along for the ride.
When casting heavily weighted flies, that weight is what you are throwing, and the fly line is going along for the ride.
Be sure to pinch down all your barbs. Because sooner or later you will bury a hook in flesh. And they come much easier when the barb is pinched down.
And yes, get a longer fly rod, like an 8 1/2 or 9 footer. That will help a great deal.