Your back and butt and core muscles are all going to have to "remember" how to flow with the horse.
The first few times I rode my horse after getting back into it after a 20year on and off relationship with horses...I thought I was going to lose all the fillings in my mouth and I won't even go into the hysterics my hubby was commenting on about my front side!
Give your body time to adjust and you'll eventually get that tush to stay put deep in the saddle!!!
Don't brace yourself against the horse...let the lower 6-8 inches of your spine "roll" with the horse. Loosen up your hips...it's a dance...Flow with your joints, don't clamp down. Pretend or actually put a $10 bill under your butt. You'll figure out how to roll instead of bounce!!! If that doesn't help, up the bill to a $20! I guarantee you'll figure it out before you hit a $100!!!!
By the way, Congrats on your new horse and Welcome to Horse Forum!!!!