You’ve answered your own question, you’re struggling with L2,R2 because you’ve got in your head that you usually just hit the button and go.
I too was a square/X user right back from Gran Turismo and other early PlayStation games, but after a time had to admit defeat that it’s just not possible to be the quickest with that configuration.
Even when the old pads had touch sensitive square and X it was still hard to use them without going to max straight away. At least the shoulder buttons give a lot more movement in comparison.
I would suggest using an auto box if you
dont already, to help try to avoid confusion when changing gear.
I made the switch with GTA5 came out, at first it felt a bit wrong but I soon picked it up and found it so much more controllable. Then when Sport came out I started with manual and wondered why I hadn’t used it in this way before, it is easily the best pad configuration!
With practice I was soon able to use Gr3 with TCS off and not be spinning out on every corner. It has shaved tons of time off my laps.
Put in some laps on time trial and persevere, you’ll be glad you did