Proper posture generally speaking is to have your head up and looking in the direction of travel, your shoulders back and down, your spine erect but not tense, and your hips square on the seat in whatever position is comfortable for you. Your arms should be parallel to your sides with your elbows at right angles and your hands held up to form a straight line from there to the bit. Keep your legs together for a lady-like (not truck driver) appearance and place one foot flat on the floor and one against the raised part of the floorboards so you can brace well. Feel free to switch which foot is forward as needed on turns.
The unfortunate thing about driving is that it is awfully hard on your back. Good suspension helps a lot and it's important to find both a well-padded seat and one whose proportions fit the driver in question. You shouldn't be leaning on the seatback unless you have to as it will effect the balance of the cart and it also tends to give your kidneys a royal drubbing when the horse gets trotting. Sit a little more forward on the seat so the seatback isn't pressing into your spine, keep your hips loose and absorbent of movement as if you were riding and breathe from the belly with the abdominal muscles taut to help keep your spine supported internally. If you're trail driving over rough ground you might also switch to open tugs instead of wrap straps and pay special attention to making sure your cart is balanced so it floats correctly. I've found that and a sliding backband saddle make a big difference! Be sure you're using breeching if you go to open tugs.
Easy entry carts are a rough ride with those seat springs and there's not that much to be done about it generally beyond the things I mention above. If you can get a cart with torsion axles or air bag suspension, do it! There's an incredible difference in the amount of jarring. Otherwise, use your joints and muscular support system as a dynamic suspension system of it's own to absorb the rattling and get out and stretch frequently on long drives.
Good luck!
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