You must keep relaxed, and have minimum tension in your body.  Even in moments we think we are relaxed, we are really tense.  Have someone watch you, and stop you every time they see your elbow bend.  You said you can do it fine standing still, so try doing it one step at a time.  Take a step and begin your pushaway, and then let the ball swing on its own.  Once you get it there, take 2 steps.  See if you can identify which step it is causing problems.  

Keep in your head that if muscling the ball was the secret to bowling, the Ah-nold Schwarzenegger's of the world would dominate.  Sometimes you need to keep these small things in mind while bowling to convince yourself to go easy.
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D <~~~~ Used to be terrible wiffing 10 pins.  Now through much practice, can wiff any single pin spare at any time.

Darrell Guzman
guzmand19 - Yahoo IM and MSN screen names

Is your grip correct? Some  experts postings having stated up to 80% of grips are incorrect. Have you seen a qualified coach? All the balls in the world can't fix bad habits or bad fit. I worked with Mike Jasneau of Lane 81 Proshop and master technician of the CATS analysis at the National bowling stadium and to me video-taping from multiple angles was quite eye opening to muscling of the swing. People can give you advice but in the end a qualified instructor is the best route.

I will say that actually learning some breathing techniques applied in my set-up and early approach did help relax my shoulders which in turn helped with some of my muscling problems. (Book from Dr Dean Hinitz- focused on bowling)