Like the old cliche' says, don't focus on swinging "faster" ... if you try to rush your swing with a heavier bat, your hips will fly open and you'll "push" your hand through the zone while the bat drags behind.
Get your new bat and a "donut" weight and start taking slow, steady swings. Focus on mechanics rather than speed. Don't just do a "warm up" swing like you'd do in the on-deck circle. Work every part of the swing every time: stride, shift then BLAST. Do three to five sets of 20 swings.
If you have a batting tee, take off the donut and hit about 50 balls off of it. Speed up to game speed, but again, focus on your mechanics and "staying behind the ball." If you find yourself pulling off the ball or hitting weakly to the opposite field, slow it down again to where you're making good contact.
With repetition, you'll build up the strength necessary to swing the heavier bat. It'll take time, so don't rush it. During the off-season, I used to take around 100 swings a night in my back yard just to keep the feel.
