I too and right handed and left eye dominant. I struggled with it mightily in the military and despite the best efforts of Navy and Marine Corps instructors never made much progress. After I got out of the service and got into guns I decided not to beat myself up about it. If I need to close one eye and shoot long guns off my left shoulder then I'm just gonna roll with it. somebody don't like it that's their problem.
I rolled like that for several decades and then one day I decided to give it another try. I have been working on it for about 5 years now and I am making progress. As far as the answer to your question on tips and tricks I guess I don't really have any. But I find that if you make the determination, be patient, and stay with it you can do better.
You may never get to the point that it feels natural or second nature. I haven't yet. But I'm doing a lot better and I expect to continue to get even better. Hang in there and stay with it. I believe you can do it. Good Luck!!
G-17, G-23, G-21, G-36, Ruger blackhawk, ss, Ruger 10/22, Ruger Mark II, Ruger SP101, British Webley MKI .455, Kel-Tec P32, S&W top break .38, S&W 24-3 .44spl N frame Colt 1903 .32, Heritage .22 Combo, Colt AR-15, LE6920.