I've given up English riding, at least until my horses are more reliable and I'm better at riding, but I'll offer some odd suggestions. You probably wouldn't want to follow them for long, but I spent two years trying without success to sit the trot, and learned in one ride how to enjoy it. Also, having someone help while I rode without stirrups wasn't an option.
First, you might find it easier to learn the feel in a western saddle. Unless your horse is a panic monster, it is pretty easy to slip your feet out of the stirrups on a western saddle, and you can use the horn to help if you start losing your balance. The stirrups are light and don't bounce much, so they don't seem to bother the horse as much as my metal English stirrups do when bouncing. And there was no way in hell I was going to try riding them with my stirrups completely unavailable...
Second, you can try homing your foot in the stirrup (putting the stirrup past the ball of your foot). When I do this, I wear cowboy boots with a 1.5" heel. I don't do it when wearing anything less. However, one of the big enemies of sitting the trot is tension in your leg. It is hard to relax the leg if you are also trying to keep your heel down. A relaxed body is far more important, IMHO, than 'core strength'...I'd bet $100 you have more than enough core strength to sit the trot.
Third, lengthen your stirrups. It may take a couple of rides, but homing your foot makes it easier to feel like you won't lose your stirrups while you get your legs longer. To break thru and feel comfortable sitting the trot, I had to practically stand with the saddle beneath me - my calf/thigh angle at about 160 deg. Any shorter, and my knee acted as a pivot point, and the horse's motion only had to lift my weight from my hips on up. Everything else was just moving horizontally to open/close my leg. More weight to go up meant I didn't move as far up.
And coming down, my relaxed legs had to spread out a bit, and that absorbed much of the shock below my hip
. My upper body cannot absorb the motion of the horse by itself. If I bend my leg too much, my thighs are too horizontal to absorb anything. They just bounce like my butt. When I get long legs (near vertical thighs) beneath me, the weight of my legs helps hold me in the saddle, and they have to move out as my hips come down to stay around the horse's barrel. My legs absorb the shock
without using pressure in the stirrups. This was critical for me!
Fourth, use zen trotting. Or trotting like an 8 year old. My daughter-in-law took her first lesson about 6 weeks ago, and sat the trot for about half of her lesson. How? I told her the most important thing was to relax and enjoy. She seemed relaxed enough that the instructor let her try trotting, and she...enjoyed it. Like a little kid does. She smiled and had fun. And sat the trot almost the entire hour the second lesson. Yes, she was tired the next day. There are muscles that need to work, but they are mostly small ones you can't control consciously anyways.
I found my conscious mind cannot do what is needed to sit the trot
. My subconscious mind can, but only if my conscious mind doesn't screw it up by tensing muscles. When I stopped trying to stop bouncing, and just enjoyed the bouncing, I started bouncing with the horse instead of out of synch. And I bounced less because relaxed muscles absorb, while tense ones reject. Try singing while sitting the trot, or whatever else it takes to get your surface mind doing something else that trying to 'help' you. It cannot.
You may look awful at first, but just keep going. Your inner mind will figure out the balance once you relax in the saddle. The horse's motion is something to enjoy, not something to avoid.
My guess is that once you learn to enjoy sitting the trot, the motion will feel natural even when you go back to an English saddle, shorter stirrups, etc. It should be fun, not work. Jack Woffard of the Shoe Bar outfit flanking the trail herd. Shoe Bar Ranch, Texas
Long leg, more forward than I would use, but the more leg below the top of the saddle, the easier I found it to sit the trot. Some of this goes contrary to what everyone else says, so feel free to totally ignore...