I was in the same situation as you, only shot .22s then bought a .270 and got the 'family' .303 at the same time. Quite a different ball game. I had a hell of a flinch to start with and was shooting worse than you are.
Lightening the trigger can help a flinch if it's too heavy to start with. I found with a heavy trigger it felt fine with no round in the chamber but as soon as I knew there was a round in there, it felt like it took a huge amount of effort to pull and it just wouldn't fire with the pull that "felt" right. The lighter trigger brought the feel of the dry fire and the live fire much closer together. All in the head, but it was much easier to get to know when it was gonna go and I could keep my trigger hand much more relaxed.
I solved my flinch with a combination of lightening the trigger on the .270 and taking my .303 (with brass butt pad) and just chucking a whole bunch of cheap rounds through it, without even really caring too much where they went at first - focusing on getting in tune with the trigger rather than the sights. It carried over to the .270 even though they're completely different in how they feel. I struggled with the scope on the .270 - I'd only ever shot with v-notch sights before and using the scope felt claustrophobic because of the recoil and jittery because of the magnification. I think that had a lot to do with my flinch. If that sounds familiar try backing off the magnification a bit (until you're used to it) so it increases the eye relief on the scope and lets you ignore a bit of jitter, and try keeping both eyes open.
Are you using hearing protection? It's better for your ears and most likely your shots, even with the suppressor.
Oddly enough, as soon as I pointed the gun at an animal rather than paper I had no issues. I wasn't thinking about the recoil or the tight groups that I wanted, so the gun felt more comfortable and easier to aim. Your results might vary, especially if you're easily excited
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