I think you may already be 'strangling it' if you're losing sight of the target on report. Having shot for far more years than I care to admit 'free-hand' the move to quad sticks was a ruddy revelation to me. Solidity 'on target' out to ranges I had previously thought as ridiculous, and a good after-shot view of the target in order to observe impact and effect. The very thought of 'making sure it doesn't move with the shot' may be why you are pulling away/flinching once the lead is airborne.
Try to set yourself in a solid but comfortable stance with the rifle on target without flexing or pushing the sticks. Squeeze slowly with the rifle firm in your shoulder and you should easily be able to watch the shot land. The old maxim of "Every time the gun goes off it should come as a surprise" tells you a lot about trigger pull and rifle-hold

Unless another bean-launcher (.308-ist) comes along to dispel this of course. It's worked fine for me so far with my .270
Damned shame you're not closer or I'd spend an hour or two down my zeroing range with you and see if I could help further.